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THERMODYNAMICS 

FOR ENGINEERS 



CAMBRIDGE UNIVERSITY PRESS 

C F CLA1, MAN unu 
LONDON FETTEELANE.E C I 




N L \\ \ ( ) R Iv. ( P l> LI T N V M ' s M ) N - 

BOMB VY 

CA.LCUTT\. M VLMILL VN VN1)( i), [ ii, 



TOfioNTo J M DFNT \NDsoNs, IIP 
TDK'S 1 1 MVRU/LN K vnUS^HIKJ Iv \ISll \ 



THEKMODYNAMICS 

FOE ENGINEEBS 



BY 



J. A. EWIJSTG, K.C.B., 
M A , LL.D., D.So., F.R.S., M tar C E , M.INST |MECH.E. 

Pimcipal and Yice-Chancelloi of the University of Edmbuigh , 

Honoiaiy Fellow of King's College, Cam budge, 

Foimeilj Piofeasoi of Mechanism and Applied 

Mechanic^ in the Univcisit) of Cambndge, 

Sometime Dneotoi of Naval Education 



CAMBRIDGE 
AT THE UNIVERSITY PRESS 

1920 



PKEFACE 

ALTHOUGH written pnmarily for engineers, it is hoped that this 
book may be of seivice to students of physics and others who 
wish to acqiuie a working knowledge of elemental y thermo- 
dynamics liom the physical standpoint. 

In piesentmg the fundamental notions of thermodynamics, 
the wnter has adopted a method which his experience as 
a teacher encomages him to think useful The notions aie 
fiist introduced in a non-mathematical foim, the readei is made 
familiar with them as physical leahties, and learns to apply 
them to piactical problems, then, and not till then, he studies 
the mathematical iclations between them. This method appeals 
to have two advantages it presents the non-mathematical 
student fiom becoming beuildeied on the thieshold, and it 
saves the mathematical student fiom am iibk of failing to 
realize the meaning of the symbols with which he plays When 
the non-mathematical student comes to face the mathematical 
relations, which he must do if he is to pass beyond the indiments 
of the subject, he finds it compaiatn ely easy to build on the 
foundation of physical concepts he has aheady laid thcie is 
perhaps no bcttei \\ay to leain the meaning and u^e of paitial 
diffeiential coefficients than by applying them to tliermo- 
dynamic ideas, once these ideas aie cleaily appiehended. 

Accoidingly the plan of the book is to begin with the 
elemental y notions and then mteipretation in piactice, and to 
defei the study of gcncial theimodynamic lelations till neai the 
2nd Finally these relations aie illustiated by applying them 
,o charactcnstic equations of fluids,, and in paiticulai to steam, 
"olio wing Callendai's method 

The chaplei on Inteinal Combustion Engines gives occasion 
or intioducmg some icsults of cxpeiiments on the internal 
'iiergy and specific heats of gases, and this matter is dealt 
vith fmther m an appendix which attempts an elementary 
iccount of the molecular theory. 

In any exposition of the first principles of theimodynamics 
t is important to choose a way of dealing with temperature 
uch that students may be led by simple and logical steps to 
nderstand the thermodynamic scale. The course followed 



vi PREFACE 

here ib first to imagine an ideal gas which serves as theimometnc 
substance, and also as the woikmg substance in a Carnot engine. 
This gives a perfect-gas scale by inference to A\hich the efficiency 
of any Carnot cycle is provisionally expressed, and from that 
the step to the thcimodynamic scale is easy 

The writei is indebted to Piofessoi Callendai and his 
publisher, Mi Edward Arnold, lor permission to include a 
much abbieviated veision of his Steam Tables. By the lecent 
publication of complete Tables, Piofessor Callendar has added 
substantial!)' to the many obligations imdei \\lncli he has 
put all students of thermodynamics. The writer would also 
thank Mi J. B Peace, of the Cambiiclge Umveisitv Press, foi 
vanous suggestions and foi the mteiest lie has taken in bunging 
out the book, and also Di E. M Hoisburgh, of the Mathematical 
Depaitment of this Univeisity, for his gieat kindness in reading 
the proofs. 

THE UNIVDRSI-H, EDINBURGH 
MaiJi 1920 



CONTENTS 



- CHAPTER I < . 

FIRST PRINCIPLES 

The Science of Thermodynamics 

Heat-Engine and Heat-Pump 

Efficiency of a Heat-Engine 

Coefficient of Peifoimance of a Refrrgeratmg Machine 

Woilang Substance 

Opeiation ot the Working Substance in a Heat-Engine 

Cycle of Operations of the Woilang Substance 

Thp Fust Law of Theimodynamics 

Internal Eneigv 

Woik done in Changes of Volume of a Fluid 

Indicatoi Diagiams 

Unit*, of Foict 1 , Pievsine, and Woik 

Units of Heat 

Mechanical Eqrmalent of Heat 

Scales of Tcmpeiatuie 

Reckoning of Tcmpeiatiue fiom the 'Absolute Zeio" 

Piopeities ot Gases Chailes' Lan and Bo\lf\ Law 

Notion of a 'Perfect" Gas 

Inter iiti! Eneigv of a Gas Joule's Law 

Specific Heats of a Gas 

Constancy ot the ^pccihc Heatb in a Peifect Gat, 

Reveisible Actions 

Adiabatic Expansion 

Isothermal Expansion 

Adiabatic Expansion of a Peitt ct Gas 

Change- of Tcmpeiatuie in the Adiabatic Expansion of a Peifect 

Gas 

Woik done in the Adiabatic Expansion of a Peitect Gas 
Isotheimal Expansion of a Peifect Gas 
Summary ot icsults foi a Perfect Gas 
Fundamental Questions of Heat-Engine Efficiency 
The Second Law of Theimodynamics 
Reversible Heat Engine Cainot's Cycle of Operations 
Carnot's Principle 

Reversrhilrty the Criterion of Petfection in a Heat-Engine 
Efficiency of a Reversible Heat-Engine 
Cainot's Cycle wrth a Peifect Gas for Working Substance 
Reversal of this Cycle 
Efficiency of Any Reveisible Engine 
Summary of the Argument 



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AST 

40 Absolute Zeio of Tempeiatme 35 

41 Conditions of Maximum Elh&iency 3G 

42 Theimodynaniic Scale of Tempeiatme 39 

43 Revcisible Engine lecemng Heat at Vaiioiva TompPiatiups i2 

44 Enhopv . 44 

45 Conseivatiou of Entiopv in Ca, mot's Cycle 45 
40 Ent;opy-Tcmpeiatuie Diagiam foi Cainot's Cycle 46 

47 Enfcropy-Tompeiatiue Diagianit, ioi a sonca of TCeveisiblc 

Engines 47 

48 JSTo change of Entiopy in Adiabatic Pioceasos 48 

49 Change of Entiopy in fin Iuo\ciaiblo Opeuition 48 

50 Sum oi the Entiopiea m a System 49 

51 Entiopy-Tempeiatuie Diagiams 50 

52 Peifect Engine using Rogoneiatoi 52 

53 Staling' a Regonpiative An Engine 53 

54 Joule's Au-Entrino 55 

CHAPTKR II 

PROPERTIES OF FLUIDb 

55 Slates of Aggregation 59 

56 Foiuiafcion of Steam undei Constant I J jcs=;iue GO 

57 iSdtuifUod and Supci healed Steam 01 

58 Relation of Piessuie to Tempeiatiuo in Satiii.itcd SI earn 02 

59 Tables of the Piopeities of titeam 62 

00 Relation of Piessiue to Volume in Sat aid ted Steam 0-1 
61 Boiling and Evapoiation 05 
G2 Mi\tiuo ol Vapoiu with othei Gases Daltou's Pniui|jU l 05 
03 Evapoiation into a space contciiinng An Hatinadon of the 

Atmospheic with Watci-Vapoiu 60 

01 Heat ipqiiued foi the Poi (nation, ot Steam undi'i C'onstant 

PHS^UIC- Heat of the Liquid and Latent Hcvifc 07 

05 Total E\teinal \Voik done 09 

06 Internal Eueigy of a Fluid 09 

07 The Total Heat" of n Fluid 70 
OS Change of the Tola! Heat dining LTenhng undei C'onst.mt 

Piessmcs 71 

69 Application to Steam foniu-d undei Constant Piessuie, horn 

WatuafcOC 71 

70 Tot\l Heat of a mixliuo ol Liquid and its Satiualed Vapom 73 

71 Total Heat of Snpei heated Vapom . 73 

72 Constancy of the Total Heat in a Tin ott ling Piocoss 74 

73 Entiopy of a PI n id 75 

74 Mi'sied Liquid and Vapom \Y<-( Steam 70 

75 Specification oi the Slato of am Flmd 77 
70 Isotheimal Expansion of a Fluid Jsofheinml Lmrs on (lie 

Piessuie-Volumo Diagiaui 78 

77 The Ciitical Point CuticaJ Tcmporadue and Cutical Piessmo 80 



CONTENTS 



IX 



ART PAGE 

78 Adiabatic Expansion of a Fluid . 8J 

79 Supersatmation 84 

80 GWngc of Internal Energy and of Total Heat in Adiabatic 

Expansion "Heat-Drop" 86 

CHAPTER III 

TIIEORV OF THE STEAM-ENGINE 

81 CdinoL'a Cycle with Steam or othei Vapoui foi Woilang 

Substance 88 

82 Efficiency ot a Perfect Steam-Engine Limits of Temperature 90 

83 Euttopy-Tomperature Diagram for a Perfect Steam-Engine 91 
Si [Jso of "Boundary Curves' in the Entropy-Temperature 

Diagram 92 

8'") Modified Cycle omitting Adrabatie Compression 94 

8f> Engine with Separate Organs 96 

87 The RanLme Cycle . 98 

88 Eihciuicy of A Rankme C^clo 99 
8 ( ) Calculation of the Heat-Drop 100 
')() The 1 Function (,' 102 
Ml Intension ot t!ie Rankine C\cle to Steam supplied in any State 104 
M2 Ro.nk.iiK Cycle with Steam initially Wet . 104 
9-} Rankine C\ do ml h Steam nutiall\ Snpeilicatcd lOb 
<n l{e\c-isibihtA of the Ilankinc CVIe 10 l ) 
M, r > Conditions ot High Efficiency 110 
Ml> Elite ( of Incomplete Evpan^ion 112 
M7 Idral Engine u 01 king \uth No Expansion 114 
MS Clapn ion's Equation ll r i 
MM \])plu.ition oi Cla,[x\ ion's Equation to utliei Changes in 

IMnsual Mali- lib 

FOiiliop^-TempLi.iluie C'hait of (he 1'iopeities ot Steam 118 

Molliei's (.'hail ot Entiopy and Total Hi.it 121 

Othoi Foi ms of Chart 125 

blik'cts of Tlnottling 120 

The Heat- Account in a Real Pioccss 129 

CHAPTER IV 
THEORY OF REFRIGERATION 

'fi Tho liofngei.ition Process 133 

(! KovciHiblo Kofiigerating Machine 134 

7 Conaeivation of Entrop3 r in a Perfect Refrigerating Piocess 135 

8 Ideal Coofhcionts of Performance 136 
[) Tlio Working Fluid in a Refngeiatmg Pioccss 137 
1 Tho Actual Cycle of a Vapour-Compression Refngeiating 

Machine 138 
1 Entropy-Tempeiature Diagiam foi the Vapoui -Compression' 

Cycle . 141 



x CONTENTS 

ART rvon 

112 Rcfngciatmg Enect and Woik of Completion, exploded 111 

Tcims of the Total Heat 144 

113 Cho,its of Total Heat and Entiopy foi Substanceb u^cd 111 the 

Vapoiu-Compiession Piocess 145 

114 Applications of the Iff) Chait in studying the Vaponi Compres- 

sion PiooebS 149 



115 Vapoui'-Compiession by means of .1 Jet W.itoi -Vapoui 

Machine 1 55 

110 The Step-down in Tcmpeiatiue Use of an Expansion Cylinclci 

in Machines using ALL 157 

117 An -Machines Joule's An -Engine leveised l r >8 

118 Dnect Application of Heat to pioduce Cold Absoiption 

Machines 30(1 

119 Limit of Ellicieney in the Use of High-tempeiatuic Heat to 

PiodnccCokr 104 

120 Expiession in Tei 111-5 ot tlie Entiop\ !()(> 

121 The Rcfiigciating Machine at, a means of \Vainniig l(>8 

122 The Attainment of Veiy Low Tempeialuie Cascade Method 1(>9 

123 Regeneiative Method 171 

124 Fust Stage 172 

125 Second Stage 175 

126 Lmde's Appaiatus J7to 

127 Liquefaction of An by Expansion in which Woik jt> done 

Claude's Appaiatus 178 

128 Sepaialion of the Constituents of An 180 

129 Ba^'sCmves 185 
1,30 Complete Rectification 188 

CHAPTER V 
JETS AND TURBINES 

131 Theoiy ot Jets 191 

132 Fonn of the Jet (De La\ al's No/7le) 103 
1M3 Limitation ot the Disehaige thiough an Oiifice of Cuen Size 197 

134 Application to An 198 

135 Application to Steam 199 

130 C'ompaiison of Mctastablc Expansion i\ith Equihbiium Ex- 

pansion 20'j 

137 Measuie of Supcisatuiation 20(> 

138 Rctaidcd Condensation 207 

139 Action of Steam in a Noz'lc, continued 208 

140 Eltects of Faction 20 ( ) 

141 Apphcaiion to Tuibineg 214 

142 Simple Tin bines 2I r > 

143 Compound Tmbine's 21 b 

144 Theoretical Efficiency Ratio 21(> 

145 Action in Successu e Stages 217 
140 Stage Efficiency and Reheat Factoi 218 



CONTENTS xi 

ART PAGE 

147 Ecal Efficiency- Ratio . 219 

148 Types of Tin bines . 220 
140 Peifoimancc of a Steam Tmbme 222 
150 Utilization of Low Piessme Steam 223 

CHAPTER VI 
INTERNAL-COMBUSTION ENGINES 

151 ' Internal Combushon . 225 

152 The Foui-Stioke Cycle . 226 

153 The Cleik 01 Two-Stioke Cycle . 226 
15 J- Ideal Action 227 
155 AH Standard 229 
15G Constant-Pie&binc Type 232 
157 Diesel Engine 234 
J5S Combustion ot Gases Moleculai Weights and Volumes 235 
151) The Giammo Molecule 01 Mol 237 
IbO The Um\eisal Gas Constant 238 
Ibl Specific Heats of Gases in Relation (o then Moletulai Weights 

VoIumcUK! Specihe Heats 239 

Ib2 Suminaiy ot Methods oi evpiessmg the Specific Heats 241 

1(53 Measui eel Values ot Specific Heats 241 

101 Vaiialion nt Specific Heat -\\ith Tempeiatuie 243 

11)5 TiUcinal Eneigy ot a Gas 245 

l(i(> Aehabatic Expansion of a Gas \\ith Vanable Specihe Heat 247 
l(>7 Ide.il Elhciency as aflccted by the Vanation ot the Specific 

Hc.il \\itliTeiupeialine 241) 

1()S ( 'line i)t liitunal KnugN toi Topical Gas Engine Mixtuic 251 
Hi!) Action in a Heal Engine Anal\si-, ot the Inclieatoi Diagiam 254 

J70 Me asuie inent ut Suction Tempeiatuie. 257 

171 The Piocess ot Explosion 257 

172 F fleet ot Tinbulvnce 259 

173 Radiation in E \plosions 2bO 
171 Moleculai Eneigy of a Gas 2bl 
175 JJisaocialion 2L>5 

CHAPTER VII 
GENERAL TIIERMODYNAMIC RELATIONS 

170 Inliodiictiou 200 

177 Functions ot Iho State oi a Fluid 206 

178 Relation of any one Function of the State to two othci & 267 

179 Encigy Equations and Relations deduced fiom them 270 

180 Expiessions foi the Specific Heats K u and Jv ;) 272 

181 Fuithcu deductions fiom the Ecaiationa foi E and 1 275 

182 The Joule-Thomson Eucct . 276 
183. Umesisted Expansion . 279 



X11 CONTENTS 



ART 



TVGE 



184 vSlopes of Lines m the 70, T$, and IP cliaita, foi any Fluid 280 

185 Application to a Jlixtuie of Liquid and Vapom in Equilibi mm 

Clapeyion's Equation Change of Phase 288 

186 Compiessibihtv and Elasticity of a JFlnicl 281 > 

187 Collected Results 2S(> 

CHAPTER VIII 
APPLICATIONS TO PARTICULAR FLUIDS 

188 Ch.uActeustic Equation 2J)0 

189 Chaiactenstic Equation of a Pcifect Gas 290 

190 IsotliPimal and Adiabatic Expansion oi Ideal Gat, 2!)2 

191 Entiopy, Eneigy, and Total Heat of Ideal Gas 20 > 

192 Ratio of Specific Heat-. Method o infemnj; y in Gasi.'a iom 

the Obseived Velocity ot Sound 203 

193 Mea-smement of y bv Adiabatic Expansion Mt'thod ot C'16- 

ineut and Desoi ines 204 

194 Eftecfc of Impei Section of the Gab on the Ratio of Specific Heats 20fi 

195 Relation of the Coohng Effects, to the Coefficients of Expansion 20b 

196 Foims of Idotheimals Diagiams oi P and V, and of PV and P 208 

197 Iinpeiiect Gat.es Amngat\ Isothennals of PV and P 20 ( ) 

198 IbOtheimals on the Picssuie-Volume Diagiain JOo 

199 Continuity ot Liquid and Gas J04 

200 Van clei Waals' Chaiacten^tic Eqmtion .{()<> 

201 Cutical Point accoiding to Van dti WaaU 1 Equation .'{O'J 
2(J2 Coiiesponding States ;j|j[ 

203 Van clei Waals 1 Equation only Appio\nnate 

204 Othei Chaiacteiibtic Equations, Claubiu-,, Lheteiici 

205 C'allendai's Equation ;jiy 

206 Deductions fiom the Callcndai Equation J21 
2U7 The Specific Heats, in C'allendai'b Equation J24 

208 The Entiopy, Eneigj , and Total Heat, m Callcnd.u\ ]^|uation ,'J2, r > 

209 Apphcation to Steam ^27 

210 Total Heat and Entiop\ ot Watei ;jjj 

211 Relation oi PiCbSine to Tempeiatuie in Satiuatecl Steam ).!( 

212 Foimukb ioi the Latent Heat ot Steam, and foi thr \ r olumc 

of a Wet Mi\tuie jjg 

213 Collected Foimulas. toi Steam yjg 

214 Tables ot the Piopei ties of tsteam jjjO 

APPENDIX I 

EFFECTS OF SURFACE TENSION ON CONDENSATION 
^ND EBULLITION 

215 Natuie ot Suiface Tension 342 

216 Need of a Nucleus gj7 

217 Kelvin's Pimciple . *,*> 

218 Ebullition 



CONTENTS 



219 
220 
22] 
222 
223 
224 



APPENDIX II 
MOLECULAR THEORY OF GASES 

PICHSIUG clue to MolecularJ'.aijja'ils 
Boyle's, Avogadro's, and Dalton's La\\ s 
Pei feet and Impeifecfc Gases 
Calculation of the Velocity of Mean Squaie 
Internal Eneigy and Specific Heat 
Energy of Vibiation 
PJanck's Formula 



PAQE 
351 
355 
350 
350 
357 
361 
3(>2 
220 Effect of Extieme Cold on the Diatomic Molecules of Hydiogen 360' 



APPENDIX III 

TABLES OF THE PROPERTIES OF STEAM 

\B1 L 

\ Piopeities of Satin ated Steam, in lelation to the Tenipeiatine 308 

A Propci ties of Watei at Satmahon Pressine 304 

B L'topeities ot Saturated Steam, in relation to the Piessme ^70 

( ' Volume ot Steam in any Diy State 372 

I) Total Heat ot Steam in any Di\ State 374 

10 Entiopv ol Steam in any Di> State 370 

I-' Specific Heat, at constant pics^uie, ot Me. am in an\ Di\ i^tato 378 

IN DUX . . . 37'J 



CHAPTER I 

FIRST PRINCIPLES 

1 The Science of Thermodynamics ticals of the relation of 
at to mechanical woik. In its engmeeimg aspect it is chiefly con- 
incd with the pioccss of getting woik clone thiough the agency of 
at Any machine i'oi doing this is called a Hcat-Engme. It is 
*a concerned with Lhc pioccss ol'icmoMiig heatfiom bodies thai" 
e cilieady coldei than their smioundings Any machine foi doing 
is is called a Refngciating Machine 

It is coinement to study the theimod> namic action of heat- 
i>incs and ichigciating machines togclhei, because one is the 
v cise of the olhi'i, and bv consideiing both \\e aimc moie easily 
an understanding ol I he whole subject. 

2 Heat-Engine and Heat-Pump In a f Icat-Enginc heat is 
pplu-d, geneially bv Ihe combustion ot luel, at a high tempcia- 
ic, mid the engine dischaigcs heat at a lo\\ci tempciatuic Thus 
a sleam-eni>ine heal is taken in at the tempciatuic ol the boilci 

id chsc'haigcd at Llie tcmpeialnic ol the condensei In any kind 

luMl-cni>mc Ihe heal is lei do\ui, \\itlim the engine, honi a high 
vcl ol leinpe'iatiiu' lo a. lowci lc\d ol Icmpciatme, and it it, by 

III ling heal down lh.it Ihe engine is, able to do woik, as a 
ilci-whccl is able to do woik b} letting \\atei doun fioni a high 
\cl to a lowei level Bui theie is this mipoitant dillcience, that 
me ol the heal disappcais in the pjoeess ol being let clown it is 
'liveried into the woik which the engine does 
In a Refrigerating Machine woik has to be spent upon the 
aclinic to enable il to lake in heat at a, low level of temperatmc, 
id dischaigc heat at a highci level of tempeiatuie, just as work 
Duld have to be spent upon a wa Lei -wheel if it were used as a 
eanb of laising watei by levcismg its action, in such a way that 
e buckets wcie filled at a low level and emptied at a higher level, 

lhat it tJiould seive as a, pump. It would be cjmte coricct to 
cak of a refuge-rating machine as a heat-pump. But again theie 
an important difference between the refrigerating machine and 

E T i 



2 THERMODYNAMICS [en. 

the r ever scdAvater- wheel the refrigerating machine is a heat-pump 
which dischaiges moie heat than it takes in, foi the work Avluch rs 
spent m driving the machine is converted into heat, Avlnch has to 
be dischaiyed at the higher le\ el of tempeiatuie in addition to the 
heat that is taken in at the IOAV tempeiatuie. 

3. Efficiency of a Heat-Engine Fiom the point of VIBAV of 
piactical theimod^namics the object of a heat-engine is to get 
Avork done with the least possible expenditure ot fuel In other 
woids the ratio of the A\ork clone to the heat taken in should be as 
laigc as is piacticable This ratio is called the Efficiency of the 
engine as a hcat-engmc The theory of heat-engines deals \\ith the 
conditions that alfect efficiency, and Autli the limit of efficiency that 
can be i cached when the conditions aie most favourable 

4. Coefficient of Performance of a Refrigerating Machine 
In a refngeiating machine Lhe object is to get heat icmovcd liom 
the cold bod}'' and pumped up to a lughei lc\ el oi tempeiatuie at 
which it can be discharged, and what is wanted is that this should 
be done with the least possible expenditure ot woik. The latio of Lhe 
heat taken in by the machine from the cold boel} to the woik that 
is spent in driving the machine is called the Cocllicient of Peifoim- 
ancc. The theory oi lefiigeiaLion deals with the conditions that 
AVill allow tint, latio to be as laige as possible 

5 Working Substance. In the action of a heat-engine or oi 
a leiiigeiatmg machine there is ah\ r a\s a woiking substance which 
foims the vehicle by which heat passes through the mac-hinc It is 
because the woiking substance has a capacity ioi taking in heat 
that it can art as a \chiclc for coin eying heat horn one level oi 
tempeiatuie to another In this process its volume changes, and it 
it, by means of changes of volume on Lhe pait of Lhe working sub- 
stance thai Lhe machine docs woik, it it is aheal-cngmc, 01 has work 
spent upon it, if it is a icfngeiating machine. Accordingly, an 
important part of the science of thermodynamics deals with the > 

piopcities of substances m i elation to heat, and the connection 
between such piopcities m any substance. The substances with 
which AVC aie chiefly concerned aie fluids in the gaseous or liquid i 

states They include an and other gases, \vater and Avater-vapour, J 

and also some fluids moie easily vaponzed than Avater, such as \ 

ammonia and carbonic acid, which aie used as the workmo sub- I 

* O f 

stance in ceitam lefng era ting machines. Each fluid has of course 



i] FIRST PRINCIPLES 3 

its own charactenstics ; but many of the relations between its pro- 

peilies are of a general kind and may be studied without limitation 

to individual fluids It will be seen, as we go on, that much of 

what has to be said applies equally, whatevei fluid selves for 

woikmg substance, and that in any one fluid the various pioperties 

are connected with one another in a -way that is true for all fluids. 

The study of the theimodynamic lelatioiibhips between the vanous 

piopeiLics of a fluid is useful, not only because of the dnect light it 

llnows on the action of heat-engines, but also because it enables 

a practically complete knowledge of the piopeities of a fluid m 

detail to be inferred fiom a compaiatn ely small number of expeii- 

menlal data We bhall see latci, for example, how such iclation- 

ships have been made use of in calculating modem tables of the 

piopeities ol steam fiom the icsults of careful measuiements, made 

in the laboiatoiy, of a \eiy few fundamental quantities. 

6 Operation of the Working Substance in a Heat-Engine. 
In oencial the uoikmg substance is a fluid which opeiates by chang- 
ing its \olumc, cxeiling piessiue as it docs so But it is easy to 
imagine a hcat-cnginc ha\uig a solid boch foi uniting substance, 
say a. long loci ol metal anangcd to act as the pa\\l of a latchet- 
whc'el wilh closely pitched teeth Lei the lod be heated so that it 
lengthens sulUciuUl\ to dine the \\heel foi \\aid thiough the space 
ol one tooth Then let the iod be cooled, sa\ bvapph ing cold water, 
llic iatchct-\\hccl being meanwhile held liom letuining by a 
separate click or detent The iod on cooling uill letiact so as to 
engage itself uith the next succeeding tooth, uhich may then be 
eh i ven foi \\ aid by heating the iod again and so on To make it 
evident that such an engine would elo woik \\e have onl) to suppose 
thai the latchct- wheel canies lound with it a chum by which a 
weight is wound up The device forms a complete heat-engine in 
which the woikmg substance is a solid iod, doing work m this case 
not through changes of \olume but thiough changes of length. 
While its length is mci easing it is exerting foice m the dnection of 
its length, ft receiver heat by being bi ought into contact with 
borne souice of heat at a compaiatively high tempeiatme, it tians- 
foims a small pail of this heat into woik, and it rejects the re- 
mainder to what we may call a icceiver of heat, which is kept at a 
compaiatively low tempeiatme. The gieatei pait of the heat may 
be said simply to pass through the engine, from the souice to the 
reccivei, becoming degraded as legaids temperature in the process 

12 



4 THERMODYNAMICS [en. 

This is typical of the action of all heat-engines, they con veil some 
heat into woik only by letting down a much largci quantity of heat 
fiom A high tempciatuic to a iclativcly lo\\ tenipeiaLurc. The 
engine we have just imagined would not be at all effiacnl, the 
fi action of the heat supplied to it which it could convcit into woik 
would IxMciysmall Much gicatci efficiency can be obtained by 
using a fluid Joi woiking subbtancc and by making it acl bo I hat 
its own expansion of volume not only docs woik bnl also can-.es it 
to fall m tcmpeiatiue bctoie it begins to lejcct heat Lo the cold 
icccivei. 

7. Cycle of Operations of the Working Substance. Ccncially 
in the action ot a heat-engine or ol a lelnaeiaLing machine the 
woiking substance ictmns periodically to the s.mic state of tcm- 
pcialuie, picssmc, volume and physical condition m all lespecls 
Each Lime this has occuncd the snbslancc is said to have passed 
thiough a complete cycle of opciations lun example, in a con- 
densing steam-engine, watci taken fiom the hoi-well is pumped 
into the boilci it then passes into the cyhnelu as si earn then liom 
the cyhndei mlo the condcnsei, and finally fiom the condense i 
back lo the hot-well, it completes I he cycle by ulunimn lo the 
same condition m all icspects as at Insl, and is ic.idv lo go Ihiough 
the cycle again In othei less ob\ ions cases a hi lie considcialion 
shows that the c\ clc is completed although lhe i same pen I ion oi 
\voikmg substance 1 elocs not go thiough il a<_>am thus in a non- 
conelensmg sleani-engme the slcam which has passed thiough the 
engine is disehaigeel into Hie atmosphcic, wheic it cools to the tcm- 
pciatuic of the feed-watt. i, while a Ciesh poilmn. ol Lccl-watci is 
deh\cie'd to Ihc engine to go llnongh the. cycle in its turn 

In the theory o( hcal-cngmcs it is of the lusl nupoi lance to con- 
sidei as a whole UK, cycle- of opeiahons pcifoinu'd by I lie 1 woiking 
substance. If we t>top shoi I of the completion of the cycle ma I leis 
aie complicated by the fact that 1hc substance is in a slate difh'icnt 
from its initial state On t he eilhei hand, if the cycle is complete we 
know that whalevci heat 01 othci energy the substance conlamul 
within itself I o begin with is tlieic still, for the state of the substance 
is the same m all respects, and consequently any woik that it has 
done must ha\c been done at the expense of heal winch il has 
taken m during the cycle The total amounl of cneigy it has paitcel 
with must be equal to the amount it lias icceivcd, duimg the cycle, 
foi its stock of internal eneigy is the same al the end as at the 



i] FIRST PRINCIPLES 5 

beginning We can al once apply I he pimciple of Ihe Conservation 
of Energy and say that for Llic cyclic process as a whole this equa- 
tion must hold good, 

Heat taken in = Heat rejected + Work done by the substance 
And sinnlaily, when the working substance in a lefngeiatmg 

machine has been cainecl through a complete cj r cle of opeiations, 

the equation holds for the c} cle as a whole, 
Heat taken in = Heat i ejected - Work spent upon the substance. 

8 The First Law of Thermodynamics. The pimciple of the 
Conscuation of Encigy m i elation to heat and work may be ex- 
pressed m the following statement, which constitutes the First Law 
of Theimodynamics When mechanical energy 'is produced from 
licat a definite quantity of heat goes out of existence fo> evety unit 
of too) L done, and, conveischj, when heat is produced by the expendi- 
ture of mechanical energy the same definite quantity of heat comes into 
existence for every unit of waiL spent 

9 Internal Energy. We have used in Ait 7 a plnase which 
icquncs some Imthci explanation the intcnial eneigy of a sub- 
stance No means exist by which the ^holc stock o eneigy that 
a substance contains can be measnicd But \\c aie concerned only 
\\ith changes in that stock, changes which ma} ausc fiom the sub- 
sLancc taking in 01 gnmg out heat, 01 doing \\oik, 01 haMiig \\ork 
spent upon it U a substance takes in heat \\ithont doing -\\oik its 
slock of mil null cncigj mciiases bv an amount equal to the heat 
taken in II it docs \\oik without lakmg m heat, it docs the woik 
at the expense ol ils slock oi internal eneigy, and the stock is 
diminished by an amount equal Lo the woik done In geneial, when 
heat is being taken in and the substance is at the same time doing 
woik, we have 

Heat taken in = "\Voik done + Inciease ot Internal Eneigy. 

Foi any mfimtcsimally small step m the pioccss, we may wnte 

dQ = dW + dE, 

wheic city is the heat taken in dming the step, dW is the woik 
done, and dE the increase of internal eneigy. 

In a complete cycle thcie is, at the end, no change of the internal 
cneigy J3, and consequently foi the cycle as a whole, 

Qi - Q. - W, 

where Qi Q.Z 1S ^e net amount of heat received, namely the 



6 THERMODYNAMICS [en. 

difference between the heal taken in and I lie heal i ejected in Lite 
complete cycle and IF is the woik done in the complete cycle 

In this notation \\c aic supposing }V to be expressed in units oi' 
heat, as well as Q and E It would be more eonect to speak oi' W 
as Hie thcimal equivalent of the woik done. 

10. Work done in Changes of Volume of a Fluid In an 
engine of the usual cyhndci and piston type the working fluid 
docs woik by changes of \olumc. The amount oi 1 ^ork done de- 
pends only on the relation of the press me to the \oliimc m lhc.se 
changes, and not on the form of the vessel 01 vessels in which I lit 
changes ot volume take place. Let the mtensit t y of prcssmc of Hie 
fluid (that it, to say the picssme on unit of aica) be P while the 
piston moves foiwaid through a small distance Sf If the aica of I he 
piston is S the total foiec on it is PS and the woik done is PM81 
But SSI = 8T, the change of \ olume hence the woik done is P&T 

iV. 

for the small change of volume 8V, or PdV foi a finite change 

r, 
of volume horn a volume T r 1 to a volume V , dining which the 

prcssme ma} 7 \ ary. 

In any complete cycle of operations the volume at the finish 

is the same as at the start, and the woik done is IPclV taken 



round the cycle a,s aAvholc 

It is vcrv useful to icpuscnt oiaplncallv I he ^\oik which .1 
fluid does in ehaiinmg its w>lumc' Lcl a diagiam be diawn m 
which the relation of the pics- 
sme of any sii])poscd woiking 
substance to its volume is shown 
by rcctangiilai cooieluuitcs as in 
fig 1. Beginning with the state jo 
rcpicscntcd by the point A, wheic f> 
the pressiuc is AM and volume 
OM, suppose the substance to 
expand to a state B, wheie the 

pressure is UN and the volume M N 

ON, and let the cm\ e AB rcprc- Volume 

sent the mtermeehatc states of ^ ^ 

prehsuie and volume. Then the work clone by the substance in this 

rON 
expansion, which is PdV, is represented bv the aiea MABN 

] OM 

under the cuive AB. 




i] FIRST PRINCIPLES 7 

Again, if the substance undergoes any complete cycle of change 
(fig 2) by expanding A 

from A through B to C 
and by being compressed 
back through D to A, 
woik is done by it while 

1) 

it is expanding from A ^ 

to C, equal Lo the aica 

MABCN, and woik is 

spent upon it while it is M N 

being compicssed fiom Volume 

C thiough D to A, equal Fl 2 

to the aiea NCDAM The net amount of woik which the 

substance does dining the cycle is equal to the algebiaic sum of 

those aicas in other woids it is equal to the area of the closed 

figuie ABCDA icprcscnting the complete cyclic opeiation, which 



aica is 



PtlV. 



If on the othci hand the operation were such as to tiace the 
fig ui c in the opposite diiechon, the substance being expanded 
fiom A to C tluongh D and compicssed fiom C to A thiough B, 
the enclosed aica would be a measiue of the \\ork expended upon 
the substance in the cycle. 

ii Indicator Diagrams This picssmc-'s olume chaoiani is an 
example, and a genciali/ation, of the method of icpicscnting woik 
which Watt iiiliodiiccd by his im ention ot the Indicatoi, an 
instalment foi automatically di awing a chagiam to icpiescnt the 
changes of prcssmc in i elation to changes of \olumc in the action 
of an engine The figuie ABCDA may be called the Indicate) 
D tag) am of the supposed action. 

The mdicatoi consists of a small cj hndei containing a piston 
which can move in it without sensible fiiction but is contiollcd by 
a stiff spi ing. This is put in free communication with one end of the 
woiking cylinder of the engine, so that the woikmg substance 
presses on the indicator piston and displaces it, against the spnng, 
thiough distances that are proportional to the piessure at eveiy 
instant. Connected with the mdicatoi piston is a pencil which rises 
or falls with it, the connection being made, geneially, thiough a 
lever that gives the movements of the indicator piston a convenient 
magnification. A sheet of paper on which the pencil maiks its 



8 THERMODYNAMICS I car. 

movements is caused to move throuqh distances piopoilional to 
the motion of the engine piston, and at light angles 1o Ihc palh oi 
the pencil Thus a dmgiam is drawn like thai ol 1 Jig. 2, exhibiting 
a closed cuive loi each double stioke ol the engine piston, .mil wilh 
cooidmates which icpicsent the changes ot piess.ne and changes 
of volume The enclosed aiea, when mtcipiclcd l>\ icleicnex to 
the appropriate scales of piessme and volume, mcasmes I he ml 
amount of woik done in the engine cyhndei dm ing [he double 
stioke, so fai as, one side of the piston is conccincd If Ihc engine 
is double-acting that is to say, if the woikmg snbsl.mcc .u-ls 
successively on the two sides of the engine pislon dm ing successive- 
strokes a similar indicator diagiam is taken foi Ihc ollui end of 
the cyhndei as well. 

12 Units of Force, Pressure, and Work Foi engine einig 
purposes, in speaking of piessme and ot woik, Hie common mill ol 
force m Bntish and Amcncnn iihagc is the weight ol 1 Ib and in 
continental usage the weight of 1 kilogiammc 1 J3y I he woid 
"weioht" we mean hcic Lhe force with which UK eailh iilh.icls 

O 

these masses When it is necessaiy to give sciuitidc puvision lo 
either of these units of foicc one must spccif\ <i locahly, 01 i.illui ,i 
latitude, because gumly acts lathci mou shongh as we no horn 
the equatoi towauls the pole. The s.uue piece ol malmal is nmie 
stiongly atliactcd b) the caith m London th.m m Pans, lo Ihe 
extent of one pait in 5000, and moic slionply in London lli.in in 
New Yoikto the extent of one pait m 1000 If llu weiulil >!' 1 II) 
of matlei m mean latitude (J5) be taken as nnily, ils weight in 
any othei latitude A is 

99735 (1 + 00053 sin 2 A) 

The differences due to lahhiele aie so small thai foi neatly nil 
pmposes they ma^ be ignoicd. 

The usual units of pitssnrc aic the pound per squ.ire me'h and llu 
kilogiammepei sqnaic ccnlinicticf Anolhei nml often used is the 
"Atmospheie," M Inch means the prc.ssme ol the almospheie ^vilh 
the baiomcter standing at 7GO mm. in latitude I. 1 } , 01 ?5!> (> mm 
m London This is equal lo a piessuic m London e>f 1 I OSO ])onnds 
pei square inch or 103274 kilogiammes pci sqnmc ccnlimelie 
Poi scientific purposes the absolute (eg s.) unit of piessme, Ihe 

* One lalogiEimino is 2 20462 Ibs 

f Sini e 1 centimetre is 393702 inch, 1 kilogramme pm iq rm is 14 22D jtoinulH 
per sq^ in , ^lien both aio measuied at the sauao place, &o Uiat gmviLy uci^ aliku on 
the pound and the kilogramme 



- L (\ 

i] FIRST PRINCIPLES ' , 9 

dyne per sq. cm., has the advantage thai it is independent," of; 
gravity. One " Atmospheie" is equal to 1'0133 x ]0 dynes peY M 
sq. cm., at any place. 

Press in es are also sometimes given in inches, 01 in millimeties, 
of mercuiy. One inch of meicury (at C ) is equivalent to 4912 
pounds pci square inch, one millunetie of meicuiy to 1 3596 
giammcs pci sq. oni. 

The usual engineering units of woik are the foot-pound and the 
mclre-kilogramme 01 kilogiammctre. One kilogiammetie is 7 233 
loot-pounds 

13. Units of Heat. For the purpose of leckonmg quantities of 
heat we compaie them with the quantity that is icquiied to warm 
a unit mass of watci fiom the tempcratme of melting ice to the 
tcmpciature at which water boils nuclei a piessuie of one atmo- 
spheie These L\\ r o points sen e to dcteimme two fixed states of 
tcmpciatuic that me quilc definite and aie independent of the 
pailiculai way in which lempeiatuie may be measmed The unit 
ol heal which i 1 - obtained by taking a ceitain iiaction of this 
riuantilyofhc.il i^ dcscnbcd as Ilie mean thermal unit The mean 
Iheiiual unit which will be used hcie is one-hunchedth pait ot the 
heat lequucd to waim one pound of \\atei lioin the melting point 
to Ihc boiling point at a piessuie ul one atmospheic Tins unit 
is called Ihe Pound-Cnlon The icason why one-hundiedth pait is 
lakcn is lliat on tlic Ccntigiadc scale of tempeiatmc the mteival 
b( Lwccn these (Kcd points is cli\ idcd into 100 degrees consequent!} 
the pound-calm > is the nvaugc amount ot heat icquiied to waim a 
pound of walci Ihiouyh one clcgice Centigiade, between the melting 
poml ,md Ihc boiling point as limits The actual amount lequued 
pci device uctd not be I he same loi each clegicc of the scale, and 
in fact is nob the same, for the specific heat ol water is not quite 
constant. 

Si milai-ly, what is commonly called the BiitishTheimalUmt (when 
the Fn.lnenh.cil scale is employed) would be defined as 1/180 of 
Ihc quantity of hcab required to waim lib of watei fiom the 
melting point to the boiling point, because on the Fahienheit scale 
there are 180 degrees between the two fixed points 

Again, the " Kilo-Caloiy " is one-hundredth of the amount of heat 
rcquucd to waim 1 kilogiamme of watei from the melting point to 
the boiling point, and the "gramme-caloiy " is one-thousandth of 
a kilo-calory 



10 THERMODYNAMICS [en. 

14. Mechanical Equivalent of Heat. The experiments of 
Joule, begun in 1843 and continued for scvcial ycais, demonstrated 
that when work is expended m producing heal a definite rcl.ihon 
holds between the amount of hcab pioducccl and Llic amounl of 
woik spent Causing the potential cneigy of a nuscd weight lo he- 
used up in tinning a paddle which generated heat b\ slnimg water 
in a vessel, and obseiving the use oi Lempcratuic so ])roducccl 5 . Joule 
made the fiist determination of the numbci ol units of woik I ha I 
aie spent in producing a mub of heat This number is called I he 
mechanical equivalent of heat. Joule found thai 77'2 fool-pounds 
weie reqmied to raise the tempeiatuie of one pound of walcr 
thiough one degiee (Fahicnheit) on the LhcimomcLci he employed, 
at a paiticulai part of the scale 

Many later and moie exact dcteiminnlions wcic made by Jonh- 
himself and by other obberveis, using vaiions mclhods of cxpui- 
ment They agree m shoeing thai Joule's oiigmal figuu' \\as 
lathei low. The general result is to fix 1400 as Lhc number ol lool- 
pounds (in the latitude of London) that are equivalent, lo oiu 
Pouncl-Caloiy as defined in Ait. 13 The coiicspondmg value 
of the mechanical equivalent of the ''Biilish Thennal UmL" is 
777 8 foot-pounds, and that of the Kilo-C'aloiv is 120 7 kilojiiaiu- 
meti es ' . 

The mechanical equivalent of heat enlcis mlo many ol Ihe loi- 
mulas of thermodynamics It is often called Joule's Equivalent, 
and is geneially represented by the symbol ,/ The symbol .1 is 
used foi the iccipiocal of Joule's eqimalenl, or ]/,/ 

15 Scales of Temperature In the constincliou of an oidin- 
aiy theimometer a fine tube of unifoim boic is chosen, and a bull) 
is formed on it to contain the mercury oi othci liquid whose expan- 
sion is to be u^ed as an indication of tempciatmc When it is filled 
the two fixed points aiedetcimmcd by placing Hie msliiimcnl (a] in 
melting ice, and (b) in the btcam coming fiom water boiling iindci 
apiessureof one atmospheic. The position lakcn by I he end ol the 
column of liquid in the tube is maikcdfoi each of (hese two points 
The distance between them is then divided into equal pai Is winch 
aie called degiees.lOOpaits foi the Centigrade scale and ISO for the 
Fahrenheit scale. By this constutcLion equal steps in tempeialuie 
aie denned by equal amounts of expansion on the pail of the 

* In absolute (c g ) units the gramme caloiy will be taken in this book IXH 
equivalent to 4 1SGS x 10 7 Digs, 01 cm dynes 



I FIRST PRINCIPLES II 

:lccted liquid, or ralhei by equal amounts of difference between 
lie expansion of Lhc liquid itself and that of the glass in which it 
. contained, for it is the diffcicncc of expansion that detcimines 
ic rise of the column in the tube This common method of 
leasurmg temperatmc gives results that vaiy for diffeient liquids 
nd foi different soits of glass Each of two mercuiy thermometers, 
n example, ma}' have the fixed points conectly marked, and be 
f unifoim boic, and -\ct if they are made of different soits of 
lass they may give leadings that diffei by as much as half a 
cgicc (Centigrade) at the middle of the lange between the fixed 
omls, and may show vciy scnons disciepancies sometimes 
mounting to as much as five degices or moic when they aie 
pphcd to measure highei tcmpciatures such as that of steam on its 
ay to an engine This illiibtiates the fact that the measmement of 
:mpeiatmc b\ an orchnaiy theunometei gives an arbitiaiy scale, 
'Inch cannot c\ en be iclied on to be the same in difleient instru- 
icnts 

Measurements of tcmpciatiue aic much less capncious if we 
~'lcct loi the expanding substance an^ one of the so-called peiman- 
nl gases, such as air, 01 mtiogcn, 01 h\diogcn, taking caie oi 
ouisc to keep I he picssine of the gas constant \\lulc it is employed 
> incasinc tcmpcialuic by its changes of volume Such an mstiu- 
icnl is called a constaut-picssuic gas thcimometei It v^oulcl be 
icoiu ciucnt loi oulinaiy use, but it sci\cs to supph a scale \\ith 
'Inch Ihc leadings of an oulinaiy Ihciniomclci can be compaied 
'luis Lhc leadings of an\ inciciny thcimometei can be collected to 
iiiii> Hum inlo agiccnicnt with the scale ol a gas theimometei if 
ha I scale be adopted a-, the standaul scale in stating tempeiatmes 

E\pciimcnls on the expansion of \anous gases by heat ha\ e 
hown lhal all gases which aie fai fiom the conditions that would 
ausc liquefaction expand vciy neatly alike Thus if we compaie an 
ir Lhci inonieler with a nilvogcn ot a hydiogen theimometei MC get 
iiaclically the same scale except at extremely low tempeiatures 
uch as those at which the gas is appi caching the liquid state, 
iascs expand by almost exactly the same amount between the 
wo fixed points, namely by 100/273 of the volume they have at the 
cmperatuic of melting ice , and at mtcimechate points, 01 at points 
icyond Lhc range, then agt cement with one another is almost 
)crfcct Hence the scale of the gas thermometer is much to be 
(referred to that of any mercury theimometer as a means of stat- 
ng temperature. But there is another and even stronger reason for 



12 THERMODYNAMICS [cn. 

this pieference We shall sec later thai it is possible 1o imagine a 
scale of tempciatuie, based on geneial theimodyiianuc pimcipks. ^ 
which does not depend on the properties of any paiticular sub- 
stance that scale is called tne thci modynamic scale of tempeiatmc, 
and much use is made of it in thei modynamic icasomng. The 
scale of a gas theimometer is piactically identical with the theimo- 
dynamic scale Taking the hydiogen theimometei, in which the 
agieement is closest, Callendar has shown ' that midway between the 
fixed points the scale collection (that is, the difference between the 
numbeis \vlucli state the same tempciatuie on the Ivydiogcn 
scale and the thermodynamic scale) is only 000] 3 of a clegicc, 
and that the tempeiatuie has to go up to about 1000 01 down 
below - 150 belbie the collection becomes as much as 1 of a ^ - 
degiee These figures are foi hydiogen expanding undei a constant J 

piessuieof oneatmospheie The differences between the scale of the 
gas theimometei and the theimodynamic scale aicc\ en less if a con- 
stants olume type of gas thermometei be used, in which increments 
of tempeiatuie aiemeasmed by the inciementsof piessmcthat are 
lequued to keep the volume of the gas constant vihile it is heated 

16 Reckoning of Temperature from the "Absolute Zero " 
Experiment shows that the amount by which an 01 ludiogcn or 
any othei so-called "permanent" gas expands between Lhe two 
fixed points that is to say in passing fioni the tempera lure of 
melting lce to that of boiling watci (at a piessmc of one atuio- j 

spheie) is about 100/273 of the volume at the lo\\ci fixed point, l 

caie being taken that the piessuie does not change Hence if ^e 
adopt the scale of the gas thermometei as our scale of tempcia- rwr 
tine, and use Centigiade dnisions, this icsult may be e\picssod ' ' 

by saying that when 273 cubic inches of gas at C aic healed 
imdei constant piessuie to 1 the \olume altcis to 27 I cubic inches 
When the gas is heated to 2C its volume becomes 275 cubic inches, 
and so on. Smulaily if the gas be cooled horn C to - ] C its 
\ olume changes fiom the onqmal273cubic inches to 272, and so on 

Putting this in a tabulai ioim, kt the volume be 
273 at C. 

It will become 272 at 1 C 



and finally would be at 273 C 

H P^ Ca I) enda ^' nt!ieAermod ^ aiulCftlc oc 
Phys boc vol svm, oc Plul Mag Januaiy, 1903 



FIRST PRINCIPLES 13 

1 the same law could be held to apply down to the lowest tempeia- 
iiies. Any actual gas would change its physical state befoie so low 

temperature weie reached, becoming fiist liquid and then solid, 
nd the volume to which it would contract would consequently be 
ot zcio but the volume of the substance in the solid state. 

The above icsult may be concisely expiessed by saying that if 
jmpeiature be reckoned not fiom the oidmaiy zeio but from a 
210 which is about 273 Centigrade degrees below it (moie exactly 
73 1), the volume of a gas, heated under constant piessure, is 
roportional to the tcmpeiatiue icckoncd from that zeio. The 
cio in question is spoken of as the Absolute Zeio ot tempeiatuie 
)enotmg any tcmpeiatiue on the oidmary scale by i and the 
oiiesponding lempeiatuie" reckoned from the absolute zeio by T, 
r c have (using Ccntigiade degrees) 

T = t + 273 1. 

The absolute zcio has been defined hcie by lefeience to the ex- 
>ansion of a gas Hut it will be been latci that the theimoch namic 
c-alc of lempciatuie blaits fiom a xeio which is absolute in the 
disc that no lowei tcmpciature can possibh exist, and that the 
eio oi I he Ihcimocl^ namic scale coincides \\ith the zeio of the gas 
calo as defined abo\c' 

17 Propeities of Gases Charles' Law and Boyle's Law. 
'he experimental fact that all pumanent" gases expand b\ veiy 
it'aily I he same ii action of then \olume toi a gn en mciease of 
empeiaLme, the pussuu- being kept constant, is known as Chailcs' 
,aw Anothei fimdamcnlal piopeih of gases, disco\eiecl by the 
xpciimcnts oi Boyle-, is thai when the volume ot a gas is alteied by 
Iteimg the picssme, the tcmpeiatiue being kept constant, the 
'olume vanes mvciscly as the picssme 

Thus if V be the volume of a given quantity of any gas, and P 
he piessmo, then so long as the temporal me remains unchanged, 
r vanes mvciscly as P, 01 PV = constant This is Boyle's Law. 
I is, vciy neaily though not exactly true in gases such as an or 
ixygen 01 mtiogen or hydrogen the deviations fiom it are very 
li<>ht m anvgas that is m conditions fai icmoved fiom those which 
)i educe liquefaction 

* Tho e\aot position of tho absolute zero is uncoitam to the extent of about 
no tenth of a degree Callendar places it at -273 1 C That figure is used in his 
oteimmations of the pioperties of steam, and is adopted in this book 



14 THERMODYNAMICS L c " 

18. Notion of a ''Perfect" Gas. In dealing with the pio- 
peities of gases and with the theimodynamics of heat-engines it is 
com enient to imagine a gas which exactly confoims to laws that 
aie only vciy neaily tiue of ical gases. Such a gas is called a 
"peifect" gas The piopcities ol'ieal gases aie mosL easily treated 
as small deviations i'lom those oi'imaginaiv "peifect" gases obe} r - 
ing simple lav\s Among ical gases hydiogen piobably comes 
neaiest to the ideal of a peifect gas., but no leal gas is in this seme 
stuctly peifect. 

In a gas which is peiiect in the sense of coiifoimmq exactly Lo 
Boyle's Law we should find PV strictly constant, so long as 
the Lcinpeiatmc is constant If we define the Umpei.ituic scale by 
refciencc to the expansion of the gas we should also have V vaiymg 
as the tempeiatuie T (icckoned fiom the absolute >icio) undei any 
constant pressinc Combining these two statements we should have 

PV = RT (1), 

wheie R is a constant 

We may wiiLe, ior any gas assumed to be pciJ'ecl, 



wheic P n and V Q aie the piessiue and volume aL C When Ihc 
volume is leekoned pei uml quantil v of Lhe gas \\L luu c a dehmte 
constant value oi R f 01 each gas, depending on Ihc units employed 
and on the specific, density of the gas in question 

It should be noticed that Avhen a gas saLislyni" this equation is 
heated nuclei constant piessme and conscqucnth expands, R i-, a 
measuie of the amount of woik done by the gas in this expansion 
for each dcgiee tlnough which the tcmpcialmc uses Let the- 
oiigmal teiu[)eiatme oi Lhe oas be r l\ and its volume V-^ and let il 
be heated undev constant piessuic P till the tcmpcialuu is '1\ and 
the \ olunie 1 T , Then we lun e R r l\ = Pl\ and Rl\_ = PV ^ horn 
which R ( r J\ - r l\] = P(V, - rj, which is Lhe woik clone In Ihc 
gas m expanding fiom V\ to F 2 Let the mlei val oftcmpeitituic be 
1, then R is equal to the woik clone. 

Thus 7? is mimevicalh expiessecl in units of woik pei unit of mass 
and per clegicc in loot-pounds pei Ib 01 in kilogiammcties [)Ci 
kilogiammc If w r c use the Centigiade degiee in both cases the 
latio of the numbei which expiesses J? in foot-pounds pei Ib to the 
numbei which expresses it in kilogiammetics per kilogramme is 
3-280S5, namely the numbei of feet in a metie 

Accoi cling to measurements by Reguault a cubic metre of dry an , 



J FIRST PRINCIPLES 15 

it a temperatme of C. and piessure of 1 atmosphere as defined 
n Ait. 12. contains 1 2928 kilogiammcs. We should accoidmgly 
lave for dry air, il it weie "peifect," 

It = I 03274 x 100 2 /1 2928 x 273 1 = 29 25, 
n kilogramme Ires per kilogiamme, at the latitude of London. The 
acLor 100- is requned to conveit the pressme into kilogrammes per 
quare metre. The coriespondmg value of R in foot-pounds per 
b. is 96-0. 

In this calculation air is Lieated as if it confoimed exactly to 
ioyle's Law For the present it is to be midei stood that the 
ymbol jT stands for tempeiaLuienieasuied on thescaleof a gasthei- 
iiomctei, liom a zeio which is 273 1 below the melting point of ice 

19 Internal Energy of a Gas: Joule's Law The Internal 
tnogy oj a given quantity of a gas depends only on the tempeiatme. 
This is an inleience fiom the fact established b} T experiments of 
ouk that when a gas expand* without doing external too? A and 
^ithoul lahing in o> giving out heat, and tliaefoie without changing 
',s -i/06'A of inteinal eneigy, its tempo ahue does not change 
Joule's Law is to be regarded as stnctly tine onh of nnaginaiy 
ciitct gases m any actual gas theic i^ a slight depaiUne liom it, 
. Inch is \ ci3 r small indeed in a nearly peiicct gas such a^> hychogen 
'he law u r as onginally established b> means oi the following 
xpeinnent. 

Joule connected a \ essel containing compressed gas uith anothei 
csscl winch was emph , In means ol a pipe \\ith a closed stop-cock 
loth vessels \veic mimeiscd in a balh ol watei and ^eic allowed to 
ssinne a unifoim temperature Then the stop-cock wat, opened, 
nd the gas distiibutcd itself betueen the t\\o \ r csscls, expanding 
ithout doing exteinal \voik Aftei Urn the tempeiatuie ol the 
r atci in the bath was found to have undcigone no appieciable 
liange The tempeiatuie of the gas appealed unalteied, and no 
eat had been taken m or given out by it, and no woik had been 
one by it. 

Since the gas had ncithei gamed nor lost heat, and had done no 
oik, its internal eneigy Avas the same at the end as at the begin- 
ing of the expciiment The piessure and volume had changed, but 
ic temperature had not. The conclusion follows that the internal 
icrgy of a given quantity of gas depends only on its temperature, 
ad not upon its pressme or volume, in other words, a change of 
ressure and volume not associated with a change of temperature 



16 



THERMODYNAMICS 



r T 



does not altei the internal energy Hence in any change of tempera- 
ture the change of internal energy is independent of the icl.ilion of 
piessure to volume duimg the operation it depends only on the 
amount by which the tempeiatuie has, been changed 

The apparatus used by Joule in thib cxpcinnenl is shown in 
fig. 3 The vessel A was filled with an compicssed to niou Ili.in 
20 atmospheres, and B was exhausted. Both \ csstls were nnmeised 
in a bath of watei The watei in the bath wns slmul and I lie 
tempeiatuie noted befoie the stop-cock C was opened Allc'i Hie 
oas had come to lest in the two vessels the water \\:is again si med, 

O 

and was found to ha\ e the same tempuatmc as bdoie, so l.u .is 
tests made by a veiy sensitive thei- 
mometei could detect. 

In another foini of the appaiatus 
Joule sepaiated the bath into three 
poi turns, one poition lound each of 
the vessels and one lound the con- 
necting pipe When the stop-cock 
wa? opened the watei suiioimding A 
vas cooled, but this was compensated 
bv a use of tempeiatuie in the uatei 
suiioundmg B and C. The gas in A became coldei 1 in llu ael <>l 
expanding, but heat was on en up in B and C'as Us eddying niolion 
settled down, and when all uas still thcie was nulliei nam noi loss 
of heat on the whole, so fai as could be delected in I Ins loim ol 
expenment. 

It is now, ho\\c\ ci, known that a very slight chani'f ol lunpc ia- 
tuie does in fact take place uhcn a ical gas expands wilhoul domn 
uoik In latei expeiimenb, by Joule and Thomson (Loid Kelvin) 
a moie delicate method was adopted of detcclmi> ulic.llui llu le is 
any change of intemaleneigy when thepiessme.ind volume cliani>< 
undei conditions such that external woik is nol done Tlie L>as was 
f oiced to pass tlnough a poioiis plug by nijiiiiliinini> <i conslanl 
liigh piessme on one side of the ping and a consUinl low pussniv 
on the other Caie was taken to picvcnl ,'inv hc.it lu-mg gnjncd 01 
lost bv conduction fiom outside In this opeiation AVOI k w.is done 
upon the gas m foicmg it up to the plug, and M'oik was done by i( 
when it passed the plug, by displacing gas undei the lower pic ssine 
on the side beyond the plug If no change of U-mpciahne look 
place, and if the gas confoimed to Boyle's Law, those two quant ilies 
of woik would be exactly equal, and consequently no e\lenial woi k 



ij FIRST PRINCIPLES It 

would be done on the whole. For IcL P be the pressure and V^ the 
volume before passing Lhe plug, and P 2 the piessure and V z the 
volume after passing the plug, the %olumes being in both cases 
slated per Jb of the gas Then the work done upon the gas (pei Ib.) 
as it appioaches the plug is PJ'i, and the woik done by it as it 
leaves the plug is P^V^ If the Lempeiature is the same on 
both sides these quantities are equal in a gas foi which PV is con- 
stant at any one tcmpeiaUue Thus a gas which is ' peifect" in the 
sense that it confoims stuctly both to Bo} le's Law and to Joule's 
would in its passage of the plug have expanded without (on the 
whole) doing any woik, and theiefoie without changing its 
internal cneigy, no heat being gained 01 lost In such a gas no 
change of tcmperatiue should accordingly be found, as it passes 
the plug, and if a change of tempeiatme is observed in any leal 
gas it is due to the fact that real gases aie not stuctly '" peifect." 

In the expcinncnts of Joule and Thomson' small changes of tem- 
peialuic wcic in lacL detected and mcasuied in an and othei leal 
gases, on passing the poious plug This Joule-Thomson effect, as it 
is called, is mgencial a. cooling Obsen ations of the Joule-Thomson 
cilVct aic of great value in dcleimimno exactly the piopeitics ol 
gases and vapouis which aic not peifeet, anel (as \\L shall sec latei) 
ceufain [uaehcal methods of hquei\mg gases undei extiemc colel 
depend upon the existence ol this c licet 

In the uuaginaiy peifect gas, ho\ve\ei, the Joule-Thomson 
ellcct is rnlnely absent Theic is no change of tempeiatuie in 
passing Ihc plug, and thcic is also no change ol internal cncigN , foi 
no woik is done and (In assumption) no heat is Liken in 01 givenout 
II is unpoitant lo notiec that \\e assume the. imatniiaiy peifect 
gas to satisfy two conditions it obc^s l$o\ Ic's Law exactly and 
also Joule\ Law exactly These chaiactciistics aie independent 
of one another it would be ])ossiblc to ha\c a gas satisiy one anel 
not the olhcj, but a gas is said to be peifect in the thcimod} namic 
sense onlv ^hcn it satisfies both, and in that case ceitam othei 
piopertics follow which will now be pointed out 

20 Specific Heats of a Gas. The Specific Heat of any sub- 
stance means the amount of heat requncd pel dcgicc to laise the 
tcnipcratuie of unit quantity of the substance, undei any assumed 
mode of heating. Thus when a substance is heated tlnough a 
small interval of tempcratuie dT the heat taken in (per Ib ) is 

* See Kelvin's Mathematical and Physical Papers, vol i, p 333 
H T 2 



18 THERMODYNAMICS i<-u 

KdT, wheie K is the specific heat for Lhe parLiculai eondilions 
and mode of heating. In dealing with gases or olhcr Hinds two 
impoitant modes of heating must be distinguished we may heal 
them undei conditions of constant piessuie 01 oi consLjuii volume 
We shall use the symbol K p to icpiesent specific beat at conslanl 
pressme, and K to repicsent specific heat at constant \ olume 

Considei first the opeiation of heating unit quantity of n peifecl 
gas at constant volume, fiom tempeiatuie r J\ up to lcm[)eialuie 
T 2 . The heat taken m it> Jr /T <. 

**-(* 2 ~ ^ !> 

No external woik is done, for the volume (by assuni])lioii) docs 
not change, and consequently all this heat goes lo muease (lie 
stock of internal eneigy contained in the gas. But by Joule's Law 
the internal eneigy depends on!) on the tcmpcratuic Thueloie if 
we heat the same quantity of the same gas in any olhei m;mnu 
fiom TI to J" 2 , the same change of mteinal cncigy must take place 
Suppose then another mannei of heating, namely at constant 
piessuie In that case the heat taken in is 

K P (T*~1\} 
Dunng this piocess external woik is done, because the gas ex- 

pands, and its amount is 

P ( V V \ 

* \ f 2 ' 1JJ 



wheie F, and F, icpiescnt the volumes al Ihe beginning {U icl end of 
the opeiation iespecti\ ely, and P is the piessme, which by assump- 
tion is constant Since PJ\ = RT 2 and PJ\ = R2\, we may \vi ilc 
the expiession foi the evteinal woik in Ihe foim 



This is in woik units in heat units it is 



wheie ^4 is the lecipiucal of Joule's equivalent (Ail 1 I) 

The dilfeience between the heat taken m and I he woik done 



is simply an addition to the stock ol mteinal enemy Jlnl 'is was 
pointed out abo\ e, the change of intcinal cncigy must be Uie same 
in both modes of heating, and theiefoie 

K v = 7v 7) AR . (>>) 

This impoitant relation between the two specific heals m a pcifecl 
gas follows fiom the Laws of Boj lc and of Joule 

We haxe heie taken K v and K, as apply mg thioni-hout a finite 



ij FIRST PRINCIPLES 19 

range of temperature from T to T 2 . But this range may be made 
mlinitesmiaJly small without affecting the argument* and in that 
case K v and K v become the specific heats at a definite tempeiature. 
The conclusion holds that for any condition of the gas 

K p ~ K v = AR, 

and this is tine whether the specific heats aie or aie not inde- 
pendent of the tempeiatme. 

21. Constancy of the Specific Heats m a_ Perfect Gas. Fiom 
the above icsult it follows that if either of the two specific heats 
is constant the othei must also be constant. To be constan 
specific heat has to be independent both of the press me ant 
the temperatme. 

Fust as to independence of pleasure- we have seen (Art 19) 
that the internal energy of a peifect gas depends only on the 
tempeiature and is independent of the piessiue. If we heat a 
peifect gas tin on oh 1 at any one tempeiatme the change of 
internal oneigy is measmed (Ail 20) by K v , no mattei what is 
the piessme. Hence K v is independent of the piessiue, and since, 
by equation (2), K v is equal to K v + AR, it follows that K p also 
must be independent of the pressure. 

But a gas may confoim to the La\\b of Boyle and Joule without 
laving K p and K v independent of the tempeiatme, and if \\e. aie 
Lo lical them as constant we must make a duthci assumption 
"cgaiding the piopeilies of that com cnicnl imagmaiv substance a 
xi feet gas Rcgnault's cxpciiments sho\\ed that in some gases /v }) is 
icaily constant tlnoui>h a modciate range of tempeiatme But it is 
low known thai in most gases the specific heat becomes distinctly 
>ieatci at high tcmpcratuics This vanation will be discussed m 
I'hapk'i VI, for oui picscnl pin pose it Avill simplify matteis to think 
>1 au ideal gas m which the specific heat is constant. Accoidmgly, 
11 dealing with a peifect gas, it is assumed that K p in such a gas is 
ti icily independent of the tempeiatme This is a thud assumed 

" Suppose the heating to bo iluough a vciy small mfceival of teiupoiaLiue dT 
n licatm^ at conslanL vohuno, the heat taken in is K dT, and all oi it goM to 
iciOdBo the intoinal oncigy bv an amount dE Hence 

K v dT=dE 

a hoatini; tiL constant piessino througli the saiae interval of temperature the 
eat taken m (dQ) does work dW and also adds to the internal energy by the 
mount dE dQ is K t ,dT, and dW is PdV, which is equal to MT Ilence 



iom winch K IS -K V =AR 

22 



20 THERMODYNAMICS [en. 

charactenstic ol a peitcct gas, additional to the two already dc- 
scubecl in Aits 18 and 19 It docs not in any way conflict uith them 
each of the thicc chaiacteiistics is independent of the otheis. With 
this finthcr assumption we have, for any pcifecL gas, K p constant 
under all conditions, and consequently K v also constant undci all 
conditions, since the diffeicnce between them is constant. 

22. Reversible Actions. We ha,\c now to consider pailiculai 
modes in which a woiking substance may be expanded 01 com- 
pressed and may take in 01 give out heat, and at the outset it is 
important to distinguish between actions that aic teveisiblc and 
those that aie n reversible 

An expansion or compression is icveisiblc if it is earned out in 
such a manner that the opeiation can be ic versed, with the lesull 1 

that the substance will pass back tluough all the stages thiongh 
winch it has passed dining the expansion 01 compicssion and be in 
the same condition in all respects at each coiiespondnig stage in 
both processes. 

This implies that the substance must expand smoothly, without 
setting up a,ivy motions within itsclL of a kind such that then kinetic 
eneigv is fntteied clown into heat through intcinal fuction The 
whnls and eddies which occui as a fluid entcis or expands in I he 
cylinder of an engine arc mcversiblc, and in ideal icvcrsiblc cxpan 
sion we must suppose them absent. Reversible expansion implies 
that there arc no losses of mechanical eflecL from any soit ol inlci- 
nal fiiction It excludes thiottlmg, such as occuis when a sub- ' 

stance expands Lhiough a valve or other constricted opening into 
a legion oi lowci press me whcic the kinetic cneigy of the sticam ' 

and eddies is dissipated. In such cases the motion of the sticam 
and eddies cannot be levcrsed. To get the subs lance back to the 
legion of highci piessurc would icqunc an expcnditme of mote 
woikthan was done upon it during its expansion, and if we wi-ie 
to loi.cc it back we should find it had gained heat tlnoiigh I he 
subsidence of the internal eddying motions, though no hc r it had 
come in iiom outside. 

The kind of expansion winch takes place in Joule's expeiimcnt 
(Ait 19) is an extreme instance o[ nievcisiblc expansion. ' 

A transfei of heat to oi from any substance is rcveisible only il 
the substance is at the same tempera tin c as the body fiom which it 
is taking heat 01 to which it is giving heat. Suppose, for instance, that 
a substance is taking in heat iiom a hot souicc and is expanding as 



ij FIRST PRINCIPLES 21 

it does so The expansion may be leversible in itself, that is to say 
it may involve no internal fnction, but unless the temperature of 
the substance be the same as that of the souice, the operation as a 
Avhole consideied in its i elation to the souice cannot be icveised. 
So consideied it is reversible only when the fuither condition is 
fulfilled that compiession of the substance willieverse the biansfer 
of heab, giving back to the source the heat that was taken from it. 
An}'' Iheimal contact between bodies at diffeient temperatures 
involves an mcversible tiausfer of heat. 

Neither the expansions and compiessions noi the tiansfers of heat 
that occiu in a ical engine aie ever stnctly reversible, some of them 
indeed aie fai fiom being levcisible. But the study of an ideal 
engine, in which all the operations aie icveisible, is of fundamental 
importance in the science of theimodynamics, and it fiunishes a 
basis foi the cntical analysis of actions in a leal engine. 

23 Adiabatic Expansion. Thcie aie two specially impoitant 
kinds of icvtisiblc c\[)ausion, (1) Adiabatic and (2) Isothcimal 

Achcibalic expansion 01 compiession means expansion or com- 
pression, earned out icveisibly, in winch no heat is allowed to 
cntci 01 leave the substance A cmve drawn to show the i elation of 
picssurc to \olume dining the piocrss is Ccilled an adiabatic line 
Adiabatic action would bcieahzcd if \\c had a substance expanding, 
01 being compiosscd, without change of chemical state, and without 
any eddying motions, m a CN lindci \\hich (along with the piston) 
was totallv imperious to heat 

Fiom Ihis definition it follows that the noik which a substance 
docb while it is expanding adiabaticallv is all done at the expense of 
its stock of inleinnl cneigy, and the \\oik which is spent upon a 
substance when it is being compiesscd achabaiicalK all goes to 
i no case its stock of internal cneigy. 

In actual heat-engines the action is never stnctlv adiabatic, foi 
Lhcie aie alwaj s some exchanges of heat between the -\\oikmg sub- 
stance and the sin face of the cylmdei and piston Very lapid com- 
piession 01 expansion may come near to being adiabatic by giving 
little time for any tiansfer of heat to occur. 

After what has been said aheady about icveisibiht}', it is 
scarcely necessary to add that expansion thiough a throttle-valve 
is not adiabatic, though it may be (and generally is) done without 
letting heat enter or leave the substance 

In the adiabatic expansion of any substance work is done, and 



22 THERMODYNAMICS [en. 

since no heat is taken in or given out, there must be a deciease of 
internal energy equivalent to the amounl of the woik done by the 
substance 
Taking the general equation (Art 9) 

tlQ = AdW + dE, 

winch applies to any small change of stale on the pait of any sub- 
stance, we have dQ, = when the action is adiabatic, and hence foi 
an adiabatic expansion 

AdW = - (IE. 

Ilcie dW is the woik done, A is the factor icqiiircd to convei t an 
expression foi woik mLo heal units (Ait 1 1), and dE is the change 
of internal eneigy 

24 Isothermal Expansion. Isolhcimnl expansion or com- 
piession means expansion 01 compression canied oul icvcisibly 
(as regaids inLcin.il aclion) and witliouL change of lempcialuie 
A cuivc diawn to show the icLition of picssine lo volume dining 
isothcimal expansion or compicssion is called .in isothennal line 

When a subslancc is expanding isolheun.illy iL lakes in heal 
to maintain ils tcmpeiatmc consLant , it Uu-icfoic mnsl be m 
con lac I wilh a souicc of heal. When il is beinn compicsscd iso- 
theimally il c;i\cs oub heat, and inusl ho in conlacL ^v r llh ,1 icceiv ci 
which can lake heal fiom it 

25 Adiabatic Expansion of a Perfect Gas. Considei nexl 
the bchavioui of a peifccl i>as dining adiabalic expansion 01 coin- 
piession. We have seen that in a small aduibalic expansion ol any 
subslnnce (Art 23) 



In a peifecL gas dE = K v dT (AiL 20) Hence in the adiabalic 
expansion of a peifcct gas 



But P = RT/r (Ail IS) Hence 

ARTdP/I r +K dT=.0, 

01, dividing by T, 

ARcW\V 4- K v clT/T = 0, 

which gives on mlcgrotion 

AR log e V + K v log c T = constant (3) 



FIRST PRINCIPLES 



23 



Writing K v ~ K v for AR (Art. 20), and dividing by K v , which 
is constant (Art. 21), 

(K,/K V - 1) log e V + log B T = constant. 
We shall wntc y for the ratio of the two specific heats, namely 



Thus we have 

y log, V - log c F + log e T = constant 
Fmlhci, since PV/T is constant, 

log, P + log e y log t T = constant 
Adding Ihese two equations 

log, P -f y log, F = constant 
which ives PF y = constant .. 



( 1). 



(5), 
(6) 



as I he equal ion of any adiabatic line in the piessiue-volume 
diagiam, to i the adiabatic expansion of a peifect gas- 1 

26 Change of Temperature m the Adiabatic Expansion 
of a Perfect Gas When a gas is expanding adiabatically its block 
cjf inUinal eiK'iy\ is, as we ha\c' seen, being icdiicc-d, and hence its 
Unipciatuie falls, the change- (jf inteinal eneigv being piopoi- 
Imiul to the change of Icnipeiatine (Ait 20) Coiuciscly, in 
adi ibahc conipiession the tempeiatnie uses The amount bv 
\\lnch the Icmpciatmc is changed (in a peilect gas) ma> be 
tound b\ r eonil)ining the equations 

i\r l v = p i vj and PJ^IP.V, - T.IT, 

Multiplying them togelhu vc ha\e 



, I , , r 

whence =- ( =- ) , 01 1 2 = 1 1 

J l V 1 

This icsult of couise applies to compiession as well as to expansion 
dong an adiabatic hue It may be got dncctly fiom equation (4), 
,vhich can be wnttcn log.T-l- (y - l)log e F= constant; whence 
Tj^- 1 - constant . (7) 

Combining equations (0) and (7) and eliminating F, we obtain 

y-\ 
he lurthci relation T/P * = constant 

* Itiatoborcmombeic(lthatlog e ,the' hyperbolic" 01 ' Napicnan" or"natuial" 
iganthm of any number, is 2 3026S5 times tho common logaiilhm of the number. 



24 THERMODYNAMICS |cn. 

27 Work done in the Adiabatic Expansion of a Perfect 
Gas. In any kind of expansion ot any fluid the ivoik done in 
expanding fiom volume J\ to \ohunc V, is 

ir = 

If the natuie of the expansion be sueh that PV n is constant, 
n being any index, then P at any point when the volume is /" is 
P 1 F 1 "/F", P 1 and F t being Ihc picssinc and \'ohinH in UK 
initial state. In that case, ioi expansion Jrom Vj lo /' ',, 

W = PJV I dF/F", 

vlncli gn es on intcgiation 

W = p Y " (V , 1 ~ n V 1 ~")/(1 n) , 

P V P V 

or jp^-A-J. ^ 2 (8) 

So fai \\ehavemadcnoassumptionas lo llic inline ol Hie \\oikmu 
substance 

Apph this icsiilt to a gas expanding adialulieallv, Ioi \\lucli 
the index n is equal to y (In Eq 0, Ail 2.;) \\ r c I hen Ii.ivi- 

W = F ^^*J^ r ^ T ^ ( c,) 

y 1 y 1 

since P 1 J r ,= li r l\ md PJ',- A"/', 

Fuithci, it follows horn Ait LM Hut llns expiession (mul- 
tiplied b\ A) is the decicasc of mlunal cncigy piodiucd b\ llu 
expansion. 

28 Isothermal Expansion of a Perfect Gas In ,i o,,s wind, 
satisfies the equation PV = JfT, Pl r is eonslanl (lining isollu iin.il 
expansion 01 completion, and an\ isolheini,il line on I he JJK-SMIK - 
volume chagiam is a lectangulai hvpcibol.i. Hie pu-ssiuc \aivin- 
imeiseh as the \ohinie 

To find the uoik done m the isolhum.il expansion of a ..as 
tiom Fj to J\\\e ha^c , 

IV = PdV 

r, 
and p = pjrjp 

fiom which ]y = ^ ir 



FIRST PRINCIPLES 25 

tegiatmg, W - P,V, (log. F 2 - log e F,), 



1 

stead of writing PjFj we may wnte PF, since the pioduct of P 
d F is constant tlnoughout the piocess, and again, since 

r = RT, V 

W = RT\o^=^ . (10), 

1 

icie T is the tcmpeiature at vhich the piocess takes place, 
is cxpiession sen es to give eithei the woik that is done by a 
3 in isotheimal expansion 01 the woik that is spent upon it in 
Lhcimal compiession 

Dining the isotheimal expansion or compiession of a perfect 
5 thcie is no change of internal eneigv, since theie is no change 
tenipciatuie and the internal eneigv depends only on the 
npeiatme (AiL 19) Hence dining isotheimal expansion a 
ifect gas must take in an amount of heat equivalent to the 
ik it does, namch JA'2 1 lot^ r,/F J9 and dining isotheimal com- 
-ssion iioin F, to J\ it must gn e out that amount of heat 



29 Summary of results for a Perfect Gas It ma\ be con- 
ucnt al llus poml to collccl the icsults that ha\ e been found 
ac Lions occiuimg in ]ieilcct gases 

it is assumed that the gas satisfies Bovle's La^ (Ait 17) and 
ulr's Law (\il 10) and that the specihc heat (at constant 
ssuie) is mdopt'ndc'iil of the tcmpciatuu Fmthci, the tem- 
aLmc is mcasmcd on llic scale tuimshcd b> the c\])ansion ot 
gas ilsclf Undci I lie-so conditions Me ha\ e the following 
I'lLs pv - RT, 

CK- 7i' is a constant depending on the specific density ot the gas , 

K^-K^AE, 

cic K v is the specific heat at constant piessuie, K v the specific 
it al constant volume and A is the icciprocal of Joule's equn a- 
l. K 1t and K v aic both constant 
n axli aba tic expansion 

PW = constant, or PjPo^ (V^JV^, 
ei c y is K V /K V 

TVy-*= constant, 01 TJT Z = (FJF^v- 1 . 

y_l 7-1 

T/P v = constant, 01 T^T,;. = (Pj/P 2 ) ? . 



20 THERMOD YNAMICh 

Heat taken in =-= 0. 

w . , R^-TJ PJ\ -PJ r , 

Work done = v l - i - 

y- 1 y- 1 

Deci ease of Internal Eneigy "^ J J 

J y \ 

In isothermal expansion 

PV constant, since T 



Heat taken m = , , 



Change of InLeinal Kncigy = 

30 Fundamental Questions of Heat-Engmc Efficiency Wt 
are now in a portion to deal \viLh MK mosl ['imdim<nl.d 
questions of htat-cngine efficiency, which m.iv In si. tied ni lh< 
following tenns 

(1) HaMng gi\en a ;,oiiicc 1'ioni ulncli lu-.il in.i> IK l.ikcn in 
at a high lempeiatiuc, and a sink 01 uci-ivi r to \\liu-li lu.il m.i\ 
be lejected at a lo\\cu tcnipcialiuo, ho\\ 111.1^ IK .il [.dun IK. in UK 
souice be utilized to the bcsLj(l\aiil,n.cloi llu |)iii|xis( (.1 |Hodiicin-. 
mechanical effect ? In olhci \\oids, Jio\\ ni.i\ (he yu.iicsl .iiii<.iin[' 
of \\oik be done b> each unit ot lit-.il l.dvi-n liom llu hu[ S.HIK, f 

(2) What fraction of the he.il Lakcn (mm (| u hoi somcc is il 
theoietically possible tocomcil into \\oiki 1 Jn olhci wt.uls, \\li.il 
is the limiting efficiency of convcisicm ? 

31. The Second Law of Thermodynamics So l.u .is llu- Fnsl 
LawofTheimodynamics(AiL S) o OCS , il isnot ol>\ ions I h.,1 (luicis 
anything to pievent all ihchenl u Inch I he somcc c.ui snppK I, on. 
being- convex Led into woik. Cut iL will j)u-si-i.[]y l sn n (h.il a 
limit is imposed as a consequence of Lhc following pmiciple uhu-h 
is known as the Second Law of Tltnmodijnan,,^ 

It u impossible foi a &clf-actn,g machinr, nnm<lr,l In, ,, nltnial 
agency, to convey heatf, om one body to unotha at a /,/;/-, /, ,, , tlltu , 

The beconcl Law says, in clfccL, that l.t-.iL will n,,l p,,ss 
automatically from a colder to a hollc-i body. \Ve can I,HC-- .[ 
to pass up, as in the action of a rcliiyc-iutmft madunc , !>( ,,| v l,x 
applying an ^exteinal agency" to dnvc the , nm | llll( .. A hr,,l- 
engine acts by lethng heat pass down from a ho, u i 1,, a oolcU- body, 
not of couise by duect conduction ftoin one Lo lh, other, l m - llu 



FIRST PRINCIPLES 27 

i mode of tiansfer in which the heat would do no woik, but by 
king the working substance alternately take in heat fiom the 
, body and icjecl heat to the cold body, and thcieby undergo 
mansions and contractions in which its pressuie is on the whole 
ater during expansion than dining contraction, with the lesult 
it a pai t of the heat that is passing down through the engine is 
i verted into work In consequence of the Second Law it is only 
eitain fraction of the whole heat supplied by the hot body that 
i be convcited mlo work by any such process 

J2. Reversible Heat-Engme. Carnot's Cycle of Opera- 

ns. To Ihc iirst ol the above two questions (Art 30) a coriect 
jwei was given by Sadi Cainot in a icmaikable essa,y s published 
1821, entitled Reflexions siu hi puissance motnce du feu et sut 

machines piopies a devcloppu edit puissance In this essay 
mot maybe said to ha\ e laid the loundation of theimodynamics 

pointed out thai the gictitest possible amount of woik was 
be' obtained by letlmg the hcdt pass fioin the souicc to the 
ci\ei tlnough an engine wuihing in a stnctly it'vo^iblc itianne) 
[ onU .is ugaids ils o\\n inlcinal actions but also as icgaids 
' hanslci ol heat to it fiom the souicc and horn it to the 
CIVLI Tlie engine conccncd by Caniot is an engine e\ci> one 
whose optialions is icvcisible in the sense explained in Ait 22. 

liuLlu'i showed how an engine might (theoietically) woik in 
h a uay as to satisfy Ibis condition, its cyele consisting of 
;sc loin icvcisible opeiations 

I) Isothciinal expansion at the tempeiatuie ol the hot souice 
L ) Duung this opeiation heat is taken in ic\ eisibly fiom the 
t soince 

("2) Adiabaiic expansion, dining which the tempeiatuie of the 
aking substance falls fiom I\ to 2 T , (the tempeiatuie of the 
eivei) 

(J) Isothcimal compiession at the tempeiatuie of the icceivei. 
irmg this opeiation heat is icjccted icveisibly to the icceivei. 
(1.) Adiabatic compiession, by which the tempeiatuie oi the 
.iking substance is raised fiom T t to 2\ This completes the 
cle by bringing the substance back to the condition in A\hich it 
is assumed to be at the beginning of the fhst opeiation. 
In the cycle as a whole work is done by the substance- the 
erage piessure in (1) and (2) being gieater than in (3) and (4). 



28 THERMODYNAMICS [en. 

This cycle of operations, which is known as Cmnot's Cycle, is 
entnely reveisible The woikmg substance might be forced lo 
go thiough it in the reveised duection, taking in heat fioin the 
cold body and gnmg out heat to the hot body The Uansfeis of 
heat would be exactly ie\eised, and aL cveiy stage the piessuie 
and volume and tempeiatmc of the substance would be Uu- s.inu 1 
as when woikmg dnect. The A\ r oik spent upon it would be equal 
to the woik got fiom it in the dnect acLion Cainol's idc.il en<>ine 
accoidingly affoids a btncllv icvcisiblc means of letting hen I 
down fiom the hot souice to the cold icccivci 

The aigument by which Carnot pioved lhat no heal-cngino can 
utilize heat moie complete!}' than a icveisible heal -engine ud!i/es 
it, in letting heat down fiom a given source lo a gn eu leceivei, 
is substantially as follows. 

33 Carnot's Principle. To prove Lhat no oLhci heat-engine, 
woikmg between the same souice and icccncr of heal, can do I IK- 
same amount of mechanical woik as a icvcisiblc engine by Liking 
in a smaller quantity of heat. 

Suppose theie aie two heat-engines 7? and <S', one o( which (A') 
is icveisible, woikmg between the same hot body 01 somei ol' lic.il 
and cold boclv 01 leceivei of heat, and each piodncmg (h r saim 
amount of mechanical Moik Let Q be Ihe qiunhh of | U<1 I \\Iuch 
R takes in fiom the hot body. Now if/? be i excised il will b v UK 
e\pendituie on it of the same amount of woik one lo UK h.[ body 
the amount of heat it foimeily took fiom il, namely Q Foi || U s 
pm pose set the engine S to diivc K incised The w.iL winch N 
pioduces is ]iist sufficient lo dine If, and Ihc I wo niaMiincs (,V 
dining R) foim togclhei a self-acling nuichmc unaidtd by any 
exteinal agencv. One of the two, nameh ,S f , lakes lual 1'iom UK- 
hot body and the othci, 7?, which u, levcisible, H i Vt s back lo UK- 
hot body the amount ol heat Q. Now if ,9 could do Us \voik by 
taking less heat than Q fiom Ihe hoi boclv ll,<> hot lm\ U W //W on 
the -Me gam heat No woik is being done on the sysU-m fiom (- 
side, nor is anj heat supplied fiom olhei somccs, so whalt x n heal 
the hot body gams must come fiom the cold body Theiefoie il ,V 
could do as much woik as the icvci sible engine If, wilh a sm illc i 
supply of heat, we should be able to anangc a pmcly sclf-ac-lmn 
machine thiough which heat would continuously pass up fmn. a 
cold body to a hot body. This would be a A u.lalion of the Second 
Law of Theimodynamics. 



FIRST PRINCIPLES 

The conclusion is that 8 cannot do the same amour 
ith a smaller supply of heat than a reversible engine , or, r 
nguagc, that no other engrne can be more efficient than a re 
igine, when they both woik between the same two temperatuJ 
source and receiver 

Further, let both engines be leveisible. Then the same argu- 
ent shows thai each cannot be moie cfficrent than the other, 
encc all rcvcrsrblc engines taking in and rejectrng heat at the 
me two temperatures are equally efficient 

34 Reversibility the Criterion of Perfection in a Heat- 
igme These results implj that, in the thermodynamrc sense, 
versibilrLy is the criterion of what may be called perfection in 
lieat-enginc A reversible heat-engine is perfect in the sense that 
cannot be improved on as regards efficiency: no other engine 
king in and rejecting heat at the same two temperatures can 
iLam from the heat lakcn in a gicalei piopoitron of mechanical 
ccL Moreover, iL this criterion be satisfied, it is, as regards efB- 
MICJ , a matter of complete mclifleiencc what is the nature ot the 
.irking subslancc 01 what, in oLhci lespecls, is the mode oi Lhe 
ginc's action 

IL is, Lheietbie, a complete answer to the fust question in Ait. 30 
say that Ihe yiealcsL amount of work that is theoretically 
issible will be done by each unit of heaL il the heat is supplied to 
L engine winch works in such a wuy thaL c\ ei^ one ot its operations 
ie\crsiblc 

35 Efficiency of a Reversible Heat-Engine The second 
ic'slion in Art .30 could not be answeied by C'ainot because in 
-, Lime lire doc-lime of the Coiisei \alioa ol Energy was unknown, 
d iL was not recogni/ed llia.1 paiL of the heat disappears, as heat, 
passing through the engine C'ainot realized that \\oik is done 

an engine through Lhe agency of heat, but he did not know that 
is done by Lire conversion ol heat It is remarkable that he was 
verlheless able to conccne the idea of a reversible engine and 
sec that rt is Lhe most effective possrble means of getting work 
lie through the agency of heat Ilrs argument as to tlm is per- 
jLly valid though rt makes no use of the JFrrst Law of Thermo- 
'namrcs It rs moreover extraordrnarrly general There rs no 
surnptron rn rt as to the proper tres of any substance, noi as to 
e nature of heat, nor as to the way in wlnnh tpmnpiatm-p is to 

measured. All that he assumes abo JISc |inn Lib B'lore 

.V 1 

/ 





THERMODYNAMICS 



[CH. 



somce and the iccenei ib Lhal one is hotter than the other The 
aigumcnt stands by itself, and the whole passage in which it is 
lepioduccd heie (Ait. 33) docs not involve a leteience to any of 
the lesults stated in caihei Aiticleb. 

But foi the puiposc of ansuenng the second question of 
Ait. 30 we shall in the fiist place deal with one particular icvcisiblc 
heat-engine, namely a ic\oihible engine which has a pcii'cct gis 
foi ^oiking substance, and shall calculate its efficiency with I he 
help of the icsulis pieviously obtained foi peifect gases. Tl will 
be easy to go on iiom that to find a general ansAvcr to the 
question, What is the hmiling efficiency of any heat-engine ? 




17')- 
Piy I- CJaiiinL'H Cycle uiih i j^as Idi woikiny 

36. Carnot's Cycle with a Perfect Gas for Working Sub- 
stance Consulci then an ideal engine in which a subslanc-c may 
go thiough Ihc opciatious ol Camel's Cycle (iig 4) [uuigine 
a cylmdu and piston composed of perfectly non-conducling 
matciial, except as i eg aids Ihc bottom oi the cyhndci, whieh is 
a conductor. Imagine also a hot body 01 indefinitely capacious 
souice of heat A, kept uh\a3"> at a Icmpcvatuic r JL\, also a pcifcc'lly 
non-conducting covci B, and a cold body 01 indefinitely capacious 
icccivci of heat C, kept ah\ay^ at some tcmpciaturc 2'_j,jjv.hicli is 
lowei than 2\. It ijs supposed that A 01 U or C can be applied at 



JfJtUJNIClPLES 31 

all to the bottom of the cylinder. Let the cylinder contain 1 Ib. 
f a peifecL gas, at tcmperatuic T 19 volume V a , and press me P a 
) begin with The suffixes icfer to the points on the mchcatoi 
lagram, fig. 4. There aie four successive operations 

(1) Apply A, and allow the piston to advance slowly through 
uy convenient distance. The gas expands isothermaJly at T a , tak- 
ig in heat fiom tlie hot source A and doing woik. The pressure 
langes to P b and the volume to V b The line ab on the indicatoi 
lagiam icpicsents this opciation. 

(2) Remo\ eA and apply B Allow Ihe piston to go on advancing, 
lie gas expands adiabatically, doing work at the expense of its 
ternal eneigy, and the tempeiatuie falls Let this go on until the 
mpciatme is T z The piessuie is then P c , and the volume V c 
us opeuition is jepicsented by the line be. 

(3) Remove B and apply C Foice the piston back slowly, 
le gas is compiesscd usothemially at T 2 , since the smallest m- 
ease of tempeiatuie abo\e T 2 causes heat to pass into C. Woik 
spent upon the gas, and heat ib i ejected to the cold icceiver C. 
t this be continued until a ceitam point ^is icached, such that 
c fourth opciation will complete the cjcle 

(I) Remove C and apply B Continue the compiession, \\luch 
now adiabatic. The picisuie and tempeiatuie use, and, if the 
mt d luis been piopcily.chosjin, when the piessuie is icstoied to 

oiii>mal value P a , the tempeiature will also ha\e usen to its 
giiul A alue r JL\ [In othci \\oids, the thud opeiation cd must be j 
ippcd when a point d is reached such that an adiabatic line chawn 

oiigh d will pass tlnough a ] This completes the cycle 

To hud the piopci place at which to stop the thud opeiation, 

have (by Ait 20), for the cooling dining the adiabatic ex- 
usion in slagc (2), 

(yjV b )y-^T b /T.= T 1 /T St 

1 also, foi the heating during the adiabatic compiession in 

- C C')' (V (l IV a )y-^T a /T^T 1 jT z . 

Icnce F c /F & =F./F aJ 

1 thcicfoic also V c jV d = V b /V a . 

That is to say, the latio of isothermal compression m the third 
gc of the cycle is to be made equal to the ratio of isothermal 
)ansion in the first stage, m older that an adiabatic line tlnough 



32 THERMODJNAJYLJLL2> |m. 

d shall complete the cycle For bie\ ity we shall dcnoU- t-illu v <>l 
these last latios (of isothermal expansion and compulsion) l>> / 
The followino aie the transfcis of heat lo and i'wm I he woilvini- 

Q 

gas, m the foui succej>si\e stages, of the cycle, quantities of heal 
aie heie expiessed in woik units 

(1) Heat taken in fiom A = m\ loy t / (I>y Ail. liS) 

(2) No Jieat taken in or rejected. 

(3) Heat rejected to C = 122' 2 log, > (by Ai I. L'S) 

(4) No heat taken in or rejected. 

Hence, the net amount of external woik done by I In- y.is, brm^ 
the excess of the heat taken m abo\c ilie heal re | a led in a com- 
plete cycle, is R (TI _ TZ ) I o ( , , 

this is the area enclosed by the loin cui\ es in Hie ligmi 
The Efficiency in this C3 r cle, namely Ihc fiac-lion 
Heat conveited into woik 
Heat taken m ' 

is aecoidmoly - 



Anothei \\ ay of bating the icsiilt is Lo sa\ lli.il il \\\ wnli Q { loi 
the heat taken in fiom the hot soiucc, and (^ loi Ihc lu'.il u |< d ( d 
to the cold leceiver, then 

QJTi - Q 3 /2' a . 

In these expie^ion-, the tempciatuics 'J\ and T, .IK iind< isl.,,,,1 
to be meabiued on the scale of a pcilcct n, ls lliuinoinclM, and 
horn the absolute zero. 

37- Reversal of this Cycle. This cycle, bcmn <t ('.H.K.I r\r!<. 
is le^eisible To icalize the fact moie lnll v we may consul,, m 
detail what will happen if we make the nnao,, 1; ,,v u>n,, l( u.,,1, 
backwaids, forcing it to trace out the same md.ralm <h.,u llim in 
the opposite oidei. Foi thi, purpose we nuisl cvp u ,d u.,,1, ,, 
it fiom some other somcc of woik Sta,l,,,o , IS brlmr |,,, IM [| 1( . 
point a (fig 4) and with the gas at I\, we shall leqnn c- 1 | u . | ( ,l!<,u ,,,,, 
toui operations. 

(1) Apply B and allow the piston to advance The o,, s , M,.,n<U 
adiabatically, the cuive tiaced is ad, and when d is reaelu 1 Ih, 
temperature has fallen to 2 1 ,. 



-j final .TJtUJ\Cll J LES 33 

(2) Remove B and apply C Allow the piston to go on advanc- 
ing. The gas expands isotheimally at T 2 , taking in heat from C, 
and the cmve dc is traced. 

(3) Remove C and apply B. Compiess the gas. The piocess 
is achabatic. The ciuve tiaccd is cb, and when b is i cached the 
tempciatme has usen to Tj. 

(4) Remove B and apply A. Continue the compiession, which 
is now isotheimal at T l Heat is now rejected to A, and the cycle 
is completed by the cuive ba. 

In this piocess the engine is not on the whole doing woik, on the 
contiaiy, a quantity of \\oik is spent upon it equal to the area of the 
diagiam, 01 R (2\ T 2 ) log t ?, and this woik is conveited into heat. 
Heat is lakcn in fiom C in the fwst opeiation, to the amount 
RTo log t ? fleat is i ejected to A in the fouith opeialion, to tlie 
amount K2\ log c j In the fiist and thud opciations theie is no 
tiansfci ol heat The machine is acting as a heat-pump 

The action is now in c\eiy icspcct the ie\eise of \\hat it \\as 
before. The substance is in Lhe same condition at coiiesponclmg 
stages in the t\\o piocesses The same woik is nou spent upon the 
engine as was loimeily done b^ it The same amount of heat is now 
given to the hot bodj A as was fonncily taken fiom it The same 
amount of heat is now laken fiom the cold bodj C as was formeily 
niven to it This will be seen by the following scheme 

Cm nofs Cycle with a peijcct gas, Dnect 

Woik done by the gas = R (T l - T 2 ] \o^ / , 
Ilcat taken liom A = RT log c ? , 
Heat icjccted to C = RT 2 log, ? 



Cycle with a perfect gas. Revei&ed 

Work spent upon the gas = R (T^ - T 2 ) log c ? , 
Heat icjected to A = R2\ log c r, 
Heat Laken fiom C = RT Z log e 1 

In the second case the heat lejected to the hot body is equal 
o the sum of the heat taken m from the cold body and the 
fork spent on the substance. This of course follows from the 
rmciple of the Conservation of Energy. 

E T 3 



31 THERMODYNAMICS |CH. 

38. Efficiency of Any Reversible Engine. The nnagmaiy 
enome, then, of Ait 36 is reveisible. Its efficiency, as we have seen, 

o y y * 

is T T 

Avhcie f l\ is thetemperatiue of the somce iiom which it takes heat 
and T 2 is the teinpeiatiue of the receiver to Avhich it i ejects heat 
But AVC saw, b\, Ait. 33, that all leveisible heat-engines taking 
m and lejectmg heat at the same tA\o tcmpciatmes aie equally 
efficient Hence the expiession 

/Tt rji 

- 1 ] ~ - 1 2 

measuies the efficiency of any icyeisible heat-engine, and theiefoic 
(by Ait. 33) also expicsses the laigest fi action of the heat supplied 
that can possibly be conveited into Avoik by any engine Avhatcvcr 
opeiating between these limits 

In other Avoids, if AVC have a supply of heat at a tempciatme 
T 1 , and a means of getting ncl of heat at a tcmpcratme T 2 , then 
theie is no possibiht_y of coin citing moic than that fraction of 
the heat into woik. Tins is the mea^inc of pojecf ejjiciauij 
it is the theoietical limit beyond which the clllcicncy of a hcat- 
engme cannot go No engine can conccuably suipass this slan- 
daid, and as, a matter of fact any ie.il engine ialK t,hoit of it, 
because no ical engine is stuctlv icxeisiblc 

39 Summary of the Argument I3ncfly iccapitulated the 
steps of the aigumcnt by which we haA r e icachcd this immensely 
nnpoitant icsult aie as follows Following Cainot, AVC considcied 
how any heat-engine woiks by taking in heat Iiom a hot somce 
and ic)ectmg heat to a cold iecen r ci, and established (by means 
of the icductio ad alisiudinn of a hypothesis Avhich Avould conflict 
Avith the Second LaAv of Thermodynamics) the conclusion that no 
engine could do this moie efficiently than a leveisible engine docs, 
that is to say, an engine \vlnch goes through a leveisible cycle ol 
opeiations Tins led to the mfeicncc that all icversible engines 
Avoiking between the same tcmpeiatuies of souicc and leceu ci AVCIC 
equally efficient, and consequently that an expression for the 
efficiency of any one of them Avould apply to all. and would mean 
the highest efficiency that is theoretically possible. Still following 
Carnot, we imagined a cycle which AAOulcl be icveisiblc, consisting 
of four stages, namely (1) isotheimal expansion dining which heat 
is taken m fiom the somce, (2) achabatic expansion dining Avhich 



FIRST PRINCIPLES 



35 



the tempciatine of the substance falls from the tempeiatme of 
he source to thai of the recen er, (3) isothermal compiession 
luimg A\lnch hcaL is i ejected to the receiver, (J<) adiabatic com- 
ncssion during \\lnch thctcmpeiatuie of the substance rises again 
o that of the source. Up Lo this point there had been no assump- 
1011 as to ihe use ol any particular working substance. We next 
nquiicd what Avould happen in this c} T cle if a peifectgas weie used 
s woikmg substance Taking loi the scale of tenipciature a scale 
kised on the expansion of a perfect gas ', and expressing on this 
eale the tcmpciatuics of souicc and reccnci as 2\ and T 2 icspec- 
ivchy, \ve found that a icvcisible engine, using a peifect gas foi 
oiking snbslancc, has an efficiency of 



[cncc it was concluded that this cxpicssion measuies the efli- 

cney ol any ie\cisible engine woikinq between these limits, and 

lat this is Llic highest efficiuic}' Iheoietically obtainable in any 

Mt-cngmc. 

This gcneial conclusion inav also be staled, with equal genei- 

ily (foi any leversiblc engine), in ihe foini 

Ci/Z 1 !-^, 

icie C^, is the heat taken m bv the engine fioni the soiuce at r l\, 
d Qy is the heat i ejected by it to the veceivci at T,. 
The efficienc of aiu liCc\L-cninc may be wiiltcn 



& Q? 

the tnginc be ie\cisi!jle 01 not In a icveisible engine, 
as we may now call it, a theiniodynamically peifect engine, 
s becomes , m ir r 

1 J. 2/J. i 

[n an engine winch falls short ol revcisibilny a smallei fiaction 
the heat supply is conveited into work and the heat rejected is 
itncly larger, Q.,/l\ is gieatci than Qi/^i 

LO Absolute Zero of Temperature. The zero from which 
and r 2 are measured is the zeio of the gas thermometer, which 
5 defined (Art. 16) as the temperature at which the volume of the 
would vanish if the same law of expansion continued to apply. 

' That IB to say, a scale in which the temperature is propoitional to the 
me ol tho gas, when the pressure is kept constant 

a a 



36 THERMODYNAMICS [en. 

But we can now give it another meaning. Taking the expression 
for the efficiency of a reveisible heat-engine 



we see that if the cold icceivei weie at the lempcialuu' ol Ihe 
absolute zeio (so that T 2 = 0) the efficiency would be equal (<> I 
mothei woids, all the heat supplied to the engine uouldbecoii\( i(< d 
intowoik It is cleaily impossible to imagine, a icceivci eoldc i lli.ui 
that, for it would make the efficiency gicalci Hum 1 and llu-u li\ 
violate the First Law of Theimoch namics by making llu amount ol 
woik done gieater than the heat supplied Hence I lie /< to which 
we found on the gas scale is also an absolute Ihcnnodvnainie /.no, 
a temperatuie so low that it is inconceivable on themiodvnaim< 
grounds that theie can be any lowci tcmpcialine The In in 
"absolute zeio" has consequently acquncd a new meaning wilh- 
out lefeience to the piopeities of any substance we see thai il 
represents a limit below which tempcialuic cannot go This 
justifies the use of the woid "absolute" as applied lo a /eio ol 
tempeiatuie. 

41. Conditions of Maximum Efficiency Funn llu .tho\< 
icsiilt it will be obvious that the avjilabilily ol lie.il loi | llin s. 
foimation into uoik depends essentially on tlu la.igr ol |,. mp , ,., 
ture thiough which the heat is let clown fiom (lul ,,| || 1( | lo( 
somce to that of the cold body into which heal is ie,,< i,.,|, ,| ls 
only in virtue of a difference of tempeiatuie be I un n bodu s I h,l 
conveision of any pait of then heat into work becomes p< IS sil,|, 
No mechanical effect could be pioduccd Irom heal, h.,w , , , ,, 
the amount of heat piesent, if all bodies wcio al a .1, .! \<d\ ,,| 
tempeiature. Again, it is impossible Lo conve.L lh, whole ol , mN 
supply of heat into woik, because it is impossible l, lux , , J,,,,|\ 
at the absolute zeio of tempeiatme as the sink mlo uhu-h ] ,l ls 
rejected. 

If T, and T t are given as the highest and lowesl I, m,,,.,,,!,,,. s 
of the lange through Mhich a heat-engine is to wmk , Is ,,, 
that the maximum of efficiency can be icached only wl,,,, |] 1( 



> * .s 



^ o f icjecliow 
Temperature of icception 



ij FIRST PRINCIPLES 37 

Any heat taken in at a temperature below T 15 or rejected at a 
tempeiature above T 2 , will be less capable of conversion into 
woik than if it had been taken in at T 1 and icjected at T a , and 
hence, with a given pan of limiting temperatures, it is essential to 
maximum efficiency that no heat be taken in by the engine except 
at the top oi the range, and no heat i ejected except at the bottom 
of the lange. Further, as we have seen in Ait. 33, when the tem- 
peiaturcs at which heat is icceived and i ejected aie assigned, an 
engine attains the maximum of eflknenc}'- if it be icveisible. 

It may be useful to repeat heie that in the tiansformation into 

kVoik of heat supplied fiom a gn en source, the condition of reveisi- 

3ihty is satisfied in the whole operation fiom souice to icceiver 

f (1) no pait of the woiking substance is bi ought into contact 

lining the operation with any body at a sensibh/ chfleient tempeia- 

ure, and (2) theic is no dissipation of eneigy thiough internal 

nction The first condition excludes any unutilized drop in tem- 

>eiatuie, the second excludes cckhing motions and such like 

oinccs ol \\asle, which ausc in consequence of expansion thiough 

hiottlc-vahcs oi coiislncted onficcs, or in consequence of any 

diisc that sets up dissipativc motions within the substance 

n a piston and cjlmdci engine we ha\e to think ol Ihe substance 

s expanding by the giadual displacement ol the piston, doing 

/oik upon it, and nol wasting eneigy to any sensible extent by 

etling poitions of itself into motion Thcie aic to be no local 

anations of picssuic within the cUmdci, such as might occui in 

fast-iimmng engine lluoui>hthc meitia of Hie expanding fluid 

When we proceed to deal in a lain ehaptei \\ith stea.ni jets in 

jlalion to steam tiubmes, \\e shall sec that it is possible toha^se 

n Llicoiy) a icvcisible action, though the woik done by the sub- 

aucc in expanding is cmplojcd to give kinetic eneigy to the 

ibslaucc itself as a whole by foimmg a jet, because in that case 

le eiioioy ol Ihc jet is icco\crable when propei caie is taken to 

mliol Ihc foimalion of the jet But the eddying motions spoken 

here aic ol a diffeienl class then eneigy is niccoverable and 

a that icason Ihcy violate the condition of icveisibihty 

It may also be woilh while to icpcat heie that no real heat- 

igme can woik between the souice and the leceiver in a strictly 

vcrsiblc mannci It cannot wholly escape eddying motions it 

nnot wholly escape tiansfcrs of heat between the woiking sub- 

ance and bodies at other tempei attires. In paiticuiai, since the 

Drkmg substance must in piactice take in heat at a reasonable 



38 THERMODYNAMICS [cu. 

rate fiom the hot somce, the source is usually much hotter than 
the substance while heat is being taken in. This is, in piactice, 
the most serious bieach of reversibility 111 the Lianslorraation of 
heat by a steam-engine. It means that between the tcmperalme of 
the soiuce and the highest temper aim e reached by the working 
substance in its cycle of operations, theie is a wasteful chop, a drop 
that is not utilized thermodynaimcally. If it were practicable 
in the steam-engine Lo avoid the drop between Ihc temperature 
of the furnace gases and that of the vsalcr in Hie boiler a gieatly 
incieased efficiency of conveision would be attainable. 

If we ka^e this diop out of account, and take for the upper 
limit 2\, not the tempeiatuie of the fuinacc gases but the temper a- 
tuie in the boiler, and if we also take for T the tempeiatuie in the 
condenser, the fraction 



will measinc the greatest fraction of the heat supplied to the 
boiler that can be conveited into woik, under idcalh favouiablc 
(in othci woids, strictly icvcrsible) conditions between the boilci 
and the condenser The perfoimancc of any ical engine falls 
shoit of this because it includes uicveisible features, the chief ot 
\\hich aic thiottling actions in the steam-passages and exchanges 
of heat between the steam and the mclal of the c} hnek'i anel piston. 
But although this limit of efficiency cannot be actually reached, 
it affords a valuable ciitcnon with which te> compaic the per- 
formance of any real engine, anel establishes an leleal J'oi engine 
designcis to aim nt 

It is important to iciih/c that a substance may expand rc- 
versibly although it is taking in heat from a souice hotter than 
rtsclf in othci woiels, thcic may be an irreversible ehop of heal 
between the somcc anel Ihc substance, but no incvcisiblc aclion 
within the snhslancc Thus Ihc fluid in a boilci is at a definite 
Lcmpcratmc lower than thai o( the lurnace while it is taking in 
heat from the fuinacc, Ihcic is accoidmgly an iiicveisiblc drop 
in this transfer e>l heal but the formation anel expansion of the 
steam may go on in a ic\eisiblc manner. We can imagine all the 
inlnnal actions of Ihc woiking substance to be icvcrsible, although 
as regaids Iransfbis of heat from the source or to the icceivci ' 

there is not reversibility. In thai event the engine will still woik 
as efficiently as possible between z/i ozou limits of lempe>atwe, JT 

namely the limits at which the substance takes in and i ejects |< 



FIRST PRINCIPLES 39 

eat, though it is no longer the most efficient possible con- 
ivance for utilizing the full range of tempera tin e fiom source to 
'ceivcr. 

Thus if we interpret 1\ and T 2 a*s the limits of tempeiatme of 

ic woikmg substance itself without any reference to a souice 

a receiver 2\ being the tempcratuie of the substance while it 

taking in heat, and T 2 the tempeiature of the substance while it 

rejecting heat, and if the internal actions of the substance are 

veiMblc, then (T l - T z )jT l still measuies the efficiency of the 

igmc. This fiaction still cxpiesses the gieatest efficiency that is 

icoictically possible in any heat-engine woikmg between the 

nits 2\ and T 2 

When we speak of a substance as taking in heat at a stated 
inpciatuic, or lejccting heat at a stated tempeiatme, it is- to be 
tderstood that the tempeiatme of the substance itself is meant, 
ough that may not be the tempcratuie of the souice 01 icceivei , 
d wjicn \\e speak of a substance as expanding 01 being com- 
cssed in a icvcisiblc mannei we do not imply that it may not be 
Ling m heat fiom a souice hottu than itsell 01 i ejecting heat to 
ccc'U ei coldei Ihan itself A c\ cle of opciatioiib ma\ be mlenuilly 
>eisible, (hat is to sa> , ic\cisibk Y so lai as actions \\ithm tlic 
uking substance aic conccincd, allhough it happens to be 
^cialcd with an me \eisiblc liansfci oi lieat to Hie woikmg 
^stance fiom the somcc oi fiom the woikma substance to the 



'CIVCl 



\2 Thermodynamic Scale of Temperature Rcfciencc was 
idem Ail 15 to the (act (lust pointed out by Loid Kch m [) that 
imodynaniic pimciplcs allow a scale ot tempeiatme to be 
uied which is independent of the piopcrlies of any paiticulai 
^stance, i-cal 01 unaginaiy Up to the piesent we ha\e based the 
le on the piopcrties of a pcifcct gas, taking a scale m uhich 
dcgtecs (01 equal mtcivals of tempciatuic) concspond to 
ml amounts ol expansion on the pait of a pciicct gas kept 
constant picssuic Using this scale we have seen that a icver- 
Ic engine which woiks between the limits T and T 2 , and takes 
any quantity oi heat Qi at T 15 icjects at T 2 a quantity Q 2 equal 
Qi T J T i> -"d has an efficiency equal to (fj - T,)/^ 

" Wo 7iiay nnagino a source at T^ and icceiver at T 2 to bo substituted for the 
al souroo and rocoivoi, it those have a wide* lango of temperatuie, without 
ting the action of the Avorking substance. 
Mathematical and Physical Papers, vol i, p 100, also pp 233 236 



40 THERMODYNAMICS [en. 

Now imagine that the heat Q 2 , which is icjccled b\ Ibis engine , 
foims the supply of a second icveisiblc engine taking in heal al T, 
and lejectmg heatatalowei tempeiatine T 3 , such lhal Hie inleival 
of tempeiatmelhiough \\hiehit woiks (T 2 - T ,) is I he same as I be 
inteival thiough which the fiist engine woiks (2\ - T,) Call e-aeh 
of these mtenals AT. Let the heat Qj icjcclcel by I Ins stcond 
engine pass on to foim the supply of a thud ie\ eisibh e ngme', \\oi k- 
mg through an equal inteival AT and icjechng heal <V, le> a loin Ib 
icveisible engine, and so on We imagine a scucs e>l engine s, e\ ei v 
one of \\hich is icversible, each passing on its ic)erle-el heal lo loi in 
the supply of the next engine in the scucs, and each vunknig 
thiough the same number of dcgiees on the peife-cl gas I he i mo- 
meter, AT. The efficiencies of the successive engines aie ^ 

AT/T 1; AT/T 2 , AT/T,, clc 
The amounts of heat supplied to them aie 



Multiply in each case the heat taken in by Lhc cllicieiiey lo Inul I IK 
amount of woik clone by each engine in the scnes, and \vi lincl lli.il 
the amount of \voik done is the same loi all the engme.s, nanu'l\ 



Accoidmgly, we might define the inteival e)l lempeialnie' loi 
each engine, without icfeience to a pcilecl gas 01 lo any ollu-i 
theimometric substance, as that inteival which makes evn\ 
engine in the senes do the same amount ol woik, and il \u- did so 
we should get a scale of tempeiatine which is ulenlical with (he- 
scale of the peifect gas thermometei. 

The above method of obtaining a llieimodynanuc scale of 
temperatme may be put thus Stalling fioni any aibihaiy con- 
dition of tempeiatine al which ucma^ imagine heal lobe supplied, 
let a series of mteivals be taken such lhal equal amounts of \\oik 
\\ill be done by e\eiy one of a seiiet, ol levcisible engines, eae-h 
woikmg with one of these mteivals foi its langc, and each handing 
on to the engine below it the heal which it icjccts, so lhal Ibr 
heat lejected by the first foims the supply of lhe> see-ond, and so 
on Then call these intervals of tcmpcialuie cenial VVh.it tin- 
above pi oof shows is that the mteivals thus ddmcd to be cepial 
are also equal when measuied on the scale ol I he pcilecl gas 
thermometer in other words, the Iheimodynamic scale and the' 



i] FIRST PRINCIPLES 41 

pcifeci gas scale coincide at eveiy point. Any tempeiatme T 
icckoncd fiom zeio on the scale of a perfect gas thermomeler is 
also an absolute tcmpeiature on the theimodynamic scale. 

The conception, then, of a chain o reversible heat-engines, each 
woikmg through a small definite lange, famishes for the statement 
of temperature a scale which is really absolute in the sense of being 
independent of all assumptions about expansion 01 othei behaviour 
of any substance As the heat goes down fiom engine to engine 
in the chain, pait of it is conveited into woik at each step, and the 
lemamdei passes on to foim the heat-supply of the next engin 
We have only to think of the steps as being such that the amount 
of heat convei Led into Avoik is the same foi each step, and that the 
remamdei passes from engine Lo engine till all is conveited. Thus 
if we have n engines in the chain, and if the whole quantity of heat 
supplied to the fiist engine is Q } , then the steps aie such that each 
engine conveits the quantity Q^jn of heat into woik When n steps 
aie completed tlicie is no heat left all is com ei ted into work 
Tins means that Ihe absolute zeio of tempeiatme has been i cached 
we may in fact define the absolute zcio as the tempeiatme \\hich 
is leached in this mannci It is imagined to be icacbed bv coming 
down tluough a JnnLc numbci of steps of tempeiatme, each step 
lepiesentmg a liiulc lall in tempeiatme We define the absolute 01 
theimodynamic scale by saving that these steps aie to be taken as 
equal to one anothei From this it will be seen that the conception 
of .in absolute 7oio, and of an absolute thermodynamic scale with 
umfoim mtcnals, docs not depend 49,11 any notion about peifect 
gases 01 about the piopeitics of any paiticulai substance We 
icaoh the absolute zeio when, on going clown tliiough the chain 
of peihcl engines, ue come to a point at which the last iiaction 
of Ihc heat has been com ei ted into woik That fixes the absolute 
zcio And we call Ihc steps by which we have come equal steps of 
tempciatuic, the steps being cletei mined by the consideiation that 
each engine in succession is to elo the same amount of woik out of 
Ihc icsuluc of heat icceivcd fiom the engine immediate^ before it 
in the sciies That fixes Ihc scale Moi cover the stops can be so 
taken, that the scale they give will agiee at two fixed points with 
the ordinary theimometiic scale, and will contain between those 
fixed points the same number of steps as the oidmary scale contains 
degiees Thus suppose the initial temperature, at the top of the 
chain, is that of the boiling point of watei, and that we have 373 
engines in the chain. Then we find that it takes 100 steps to come 



-it THERMODYNAMICS [en. 

do Mil to the temperature of melting ice, and 273 moie steps 1 to 
complete the convcision of the remaining heat into work. This 
means that the uniform step of tempcialuic on, the thcimodjnamic 
scale is equal to the a\ eiage of the mtcivals called dcgiccs on any 
centigiade theimometei, when that average is taken between the 
freezing point and the boiling point (0 and 100), and that the 
absolute zeio is at a point 273 of such steps' ijeJow the freezing 
point But the llicimodynamic scale Mould agice from point to 
point vi Hi the indications of the thcrmomclci throughout: the whole 
of the scale oiuy if the theimomeler could use a peifect gas as ils 
expanding substance Even with h}diogen, which is ^ery ncaily 
a peifect gtiSj theie are slight cliveigenccs which \\eic mentioned 
in Ait 15. 

43. Reversible Engine receiving Heat at Various Tempera- 
tures. In Camel's cycle it was Assumed that thcic was only one 
soiucc and one reccuei of heat All the heat thai was taken in 
was taken in at2\, all the heat that was rejected was, icjectcd atT 2 
But an engine may take in heat in slaves, at moic tempciatuics 
than one, and may also icject heat in stages Wilh j eg aid to 
cvciy quantity of heat so taken in, the result still applies that I he 
greatest fraction of it that can be converted into woik is icpic- 
senlcd by I lie diffciencc between its tcmpeiatuics of rcccptioii and 
icjcclion, dnielcd by the absolute lempciatuic of icceplion And 
this is the 1'iaclion that will be coiucitid uilo work piovided the 
pioccsses within the engine aic icvci->iblc. 

Thus if Q z represents that paitol' Ihe whole suppb/of heal which 
is taken in at r J\ and Q, icpiescnts what is lakcn in at some olhci 
lempciatuic To, Qj at T 3 , and so on, and if T () be the tcmpeialuie 
at \\luch Ihc engine icjccls heal, UK whole \\oik done, il Lhc 
engine iti icversible, is 

ft (T, - T ) <2 2 (T, - T ) Q, (T, - T {] ) 
fr - ,,, -| ,-- -t- - ~,., r ci,c. 

J 1 ^2 J A 

We heic take, foi simplicity of statement, a single tcmpeialuie of 
rejection T 

A mechanically analogous machine \\ r ould l)c a gicat walei- 
Avhccl, working by gravity, and iccciving water mlo its buckets 
fiom leseivons at various levels, some of which aic lower than the 
top of the wheel. Let Jl/ a , M z and so on be the weights of water 

* Moie exactly 273 and a nation (Ait 10) 



FIRST PRINCIPLES 43 

ccived at heights I l3 1 2 etc. above any datum level, and let 1 
1 the height above the same datum level at which the water 
ives Ihe wheel If the wheel is peifecLly efficient (and hcie again 
c test of pa feet efficiency is re vei ability) the work done is 

MI (h ~ k) + M z (1 2 - 1 ) + IT, (1 3 - Z ) + etc. 

mipanng the two cases we sec that the quantity Q.JT l is the 
aloguc in the heat-engine of l/ x in the watei-wheel, Q 2 [T Z is 
<- analogue ol AI 2 , and so on. The amount of work which can 
got out of a <> i ven quantity of heat bjMeJttmg.ji_dg_w_n to an 
" ~l !~ '" ..',. '""V 1 ,, . '. ' '-..the 

(..!.i 'i . v ( , I. , 'i . L . i ; ! t to_ 



e piodnct ofJ2/TJby_ Lli^faJI^LtcinpciaJuiUL On the stiength of 
is analogy Zcunei has called the quantity Q/T the ''heat weight" 
a quantih ol heat Q obtainable at a tempciatuie T 
AnoLhei way ol' pulling Hie inaltci has a \\idei application. 
t the engine as beloie lake in quantities of heal icpicsented by 
,^,Q,i'lc at^V/Vy'^andletil iqtct heal at T',T",T'" etc , 
1 quantities u'jectcd l)cin<> ic'spcch\ el\ Q',Q",Q'"cic Tlan by 
1 pnncipk thai in a ie\eisible cycle Hit heal u (ected is to the 
il laken in as the absolute lempeialuu ol ujeclion is to the 
-.olutc kinpeiahiu of icceplion, nc have 

Q' , ^2" , '" , _ i , ft , Q* , 

T' T" T" " 2', T 2 T 3 
^Q. 

m Avluch S = 0, 

en the simimahon is clTeeled nil lound Lhc ie\cisible e\clc 
llus suinmalion heal taken in is icekoned as ])osili\c and heat 

I'ck'd as negalivc If (he c\ele is not icversibk, the heat ic- 

Icd will be iclatively gicatci, and lliuefoic, foi a non-ieversible 

le, H (Q./T) will be a negative quantity 

ionic of the pioccsscs may be such that changes of tcmpeiatuie 
going on continuously while heal is being taken in or given out, 
1 if so we cannot divide the icceplion 01 lejection of heat into 
nulcd number of steps, as has been clone above. But the equa- 
i may be adapted to the most geneial case by wntmg it 



sgration being performed round the whole cycle. 

This holds for any internally rcvei sible cycle It means that when 



44 THERMODYNAMICS | < 1 1 

a substance has passed through any senes of levoisible ohangis 
which cause it to leturn to its initial stale, the (pi.inlilns ol IK a I 
which it has taken in and given out aie so related lo UK lim- 
peratme of the substance at each stage as to make this mlcgi.il 
vanish foi the cycle as a whole If the cycle is not reveisible 
jdQ/T is a negative quantity, because the amount of IK a! 
icjected is relative^ laigei than when the cycle is icu-isihlc 

44 Entropy. We have now to inlioduce an impoitanl Iheimo- 
dvnamic quantity vilnch serves many useful ])ui pose s The ttntio/>i/ 
of a substance is a function of its state which is most COIIM UK nll\ 
denned by icference to the heat taken in or gixen out x\h< n UK 
state of the substance undeigoes change in a icveisibk maniK i 
In any such change, the heat taken in or gnen out, divided b\ 
the absolute tempeiatuie of the substance, measmes I he- ehang< ol 
entropy Thus if a substance which is cilhtr e'X[)andmg n \ eisiblv 
or not expanding at all takes in heat 8^ wJien its le-mpi i.ilnx is 
T, its entropy mci eases by the amount 8Q/7 1 We shall see Ilia I UK 
entiopy of any substance in a definite slate is a delimit- <punlil\, 
which has the same value when the substance comes bae-k again lo 
the same state after undeigomg an> changes To gi\ e I h, e nl i ops 
a numerical value we must stait lion, some aibiliaiy p,.n,l uh, i, 
foi convenience of leckonmg, the entiopy is taken as /e,o \\ ,' 
aie conceined only with changes of entiopy and rons,,,,,, mix ,1 
does not mattei, except foi convenience, what /c.o stale is H,,,,, 
toi the purpose of calculating the entiopy 

Starting then from any suitable zero,' each clement *Q ol UK 
heat taken in has to be divided by T, which ,s the absol,,| ( ,, ,, u . 
tuie of the substance when 8Q is being taken m Tin 



sum 



the enhopy of the subsUnoo, ,1,, ....... , ........ , , 

W, T'f " Se f StatC haS cc " ri ' 1K M- , ..... J 

la shall denote the ent.opy of v s,,bs llu ,. |, v ,/? , , , 

pe.ature chan s mo oont,no,,slj. wh.le henl ls I,, ', ,.,! , , 
change of enhopj- fiom any state lo an, o.iu,. s |.u,' 7," 



mevasibie 



MJttoT JfKINCIPLES "' 45 



This definition of the entiopy of a substance as a quantity 

f 6 dO 
uch is to be measured by reckoning -^ while the substance 

a * 

sses by a reveisiblc process from any state a to any other state 
is consistent with the fact that the entiopy is a definite function 
the state of the substance, which means that it has only one 
ssible value so long as the substance is m the same state. To 
jve this we must show that the same value is obtained for the 
liopy no mallei what icvcisible operation be followed m passing 

6 ,7/~l 



m one slate to I he othci in othei woids, thai 



f is the same 



a 



all icveisiblc opciations by which a substance might pass horn 
Ic a to state b Considci any two icveisible ways of passing 
in stale a lo state b If \\c suppose one of them to be icversed 
i two toi>ethci will foim a complete cycle foi which (by Ait 43) 

= He nee I r for one ol them iniibt be the same as 
t (l J. 

the oihci il is Ihcicfoica inattei oi'indiflcience, in the icckon- 
,of entiopy, by \\haL ' path" 01 scqucnccoi cliangcs the substance 
>scs liom a to b pioudcd it be a icvcisiblc path staitmg fiorn 
y /cio slalc the leckonmfi of the enliopy in a gixui slate \\ill 
k r ays gi\e Llie same \alue, winch shows that the entiopy is 
iply a lunction of the actual state and docs not depend on 
'vious condihons 

[I is chiefly because the entiopy of a substance is a definite 
iction of the stale, like the leinpuatuic, 01 the pic&smc, 01 the 
umc, 01 the internal eneigy, that the notion of entiopy it, nn- 
tant in cngmecung Ihcoiy The entiopv of a substance is 
idlly icckoned pei unit of rnasb, and numencal values of it 
koncd in this mannci aie given in tables of the pioperties of 
am and of the othei substances which aie ut,ed m heat-engines 
1 refiigciatmg machines. 

But we may albO leckon the entiopy of a body as a whole when 
state ol the body is fully known, or the change of entiopy which 
>ody undciyocs as a whole when it takes in or gives out heat 
d we may also leckon the total entropy of a s} r stem of bodies 
adding togethei the entiopies of the seveial bodies that make 
the system 

5. Conservation of Entropy m Carnot's Cycle. As a simple 
stration of the uses to which the idea of entropy may be put, 



* THERMODYNAMICS [cn 

considei the changes of enLiopy which a substance undeigocs 
when it is taken through Cainol's cycle (ArL. 32). All fom opera- 
tions are ie\ei'sible. In the fust, which is isoLhcimal expansion 
at T lt the entropy of the substance mci eases by the amount QJTj^ 
where Q a is the amount of heat taken in from the hot source. 
In the second operation no heat is taken in 01 given out and there 
is no change of cntiopy In the thud opciation a quantity of 
heat Qa ls rc]cctcd at T z the cntiopy of the substance accoidingly 
falls by the amount <2 2 /T 2 J- 11 tae fourth opeiation theic is again 
no transfer of heat and no change of cntrop}'. It is only in the fiisL 
and third operations that changes of enLiopy occui. Moicovcr 
they aie equal, foi Q l j2\=^ QJT 2 , which shows thai the substance 
has the same cntiopy as at fust, when it has icLuincd to the 
oiigmal state 

During the lii^t opeiation, while it was talcing in heat, its entropy 
rose fioni the initial value, which we may call (j> a , to a \aluc (j> b 
such that / / , /i in-, 

</>& = </> + QiM i 

Dining the thud operation, while the substance was rejecting 
heat, its entiopv t'cll again from cj) b to (/>, and 

ta - fa ~ W* 

Taking the C3'cle as a whole, the thermal equivalent of the 
woik done bv the substance is Qj Q.*, <i'id is accordingly equal to 
(T l - T 2 ] (^ ~ a ). 

Furlhci, the source of licat has losl ,m amount of entropy 
equal to Q. i jT i , and the receiver has gamed an equal amount of 
entrop\, namely Qj/jT, We may thercioie regard the reversible 
engine of Car not as a device winch Lransleis cntiopy Jiom tire 
hot source to I he cold receiver without altering the amount ol 
the cntiopy so tiansfeired The amount of heat altert> in Ihe 
process of UaiihCci, for an amount of heat Q t Q, disajjpeais, winch 
is the thermal equivalent of the work done, but the amount of 
entropy rn the system as a whole does noL change 

If, on the other hand, we had to do with an engine winch is 
not reversible, uorking between the same source and recerver, Q n 
would be relatrvclv larger, since less of the heat taken in is con- 
verted nrto work Hence Q 2 /T 2 would be greatei than Qj/T t and 
the amount of entropy would theiefore increase in the transfer 

46. Entropy-Temperature Diagram for Carnot's Cycle. It 

is instructive to represent the changes of entropy rn a Carnot 



FIRST PRINCIPLES 



m 

Entropy 
Vis 5 



cle by means of a diagram the two coordinates of which are the 
tropy of the woikmg substance and its tempeiatmc (fig. 5) 
,e first operation (isotheimal expansion) is icpresented by ab, 
.tiaight line diawn at the level of temperatme T l . dining this 
jiation the cntiopy of the substance uses from </> a to <^ 6 This 
followed by adiabatic expansion be during which the tem- 
atuic falls but the entropy does not change Then it>othei- 
1 compicssion cd at tempeia- 

c 7'o,duung which the entiopy , 

uinstolhcimtialvalue Finally 
aba I ic compression da com- 
ics the cycle j 

The aica of the closed figuie -5 

d c 

d mcasuics (m heat units) the <u 

ik done duimg the cycle The (= 

a niabn mcasmcd to the base L 

', which is the absolute zcio of 

ipciaturc, is the heat taken m 

n I he SOUK c The aiea indui is 

heat ic|ccled to Ihciccenci 

se (igmes aic icclanglcs 

,11 [his is liue^\luitcvci be the \\oikmg substance Neither m 
15 iioi lii-ii- is ,iii> assumption made as to that The diagram 
5) applies to any engine going thiough the icvcisiblc cycle of 

not win UK i it use a gas (as in Ail 30) 01 anv othei substance 

7 Entropy-Temperature Diagrams for a series of Rever- 

e Engines We may apply this 

hod ol lepiesenUilion to exhibit 

action ol the imaginary chain of 

isible engines which was used 

Lit 12 to establish a thermody- 

ue scale of temperature. 

artmg fiom any temperatme 

et a icversible engine take in 

at that temperature, and go i 

ugh the Cainot cycle of opera- 

> lepiesentcd by the icctangle 

For this purpose it takes m 

equivalent to mdbn and i ejects 

equivalent to mdcn. Let its 

ted heat pass on to the next 




1-u X ilJii J.X1U.VJ LJ i IN AlU J. V,O [ CIJt ' 

engine of the series, which goes through the Carnot cycle dccf, 
and let the inteival of tempeiatuie rf/be so chosen as to make the ^ 

woik done by the second engine equal to the woik done by the / 

fiist. Fiom the geometry of the figme it is obvious that this 
requncs ilf to be equal to ad, so Lliat the area abed may be equal 
to the aiea dccf. Sumlail} in oidci that the woik clone by the 
thud engine should be the same, we must have fh -= df ad, and 
so on. Thus these mteivals constitute equal steps in a scale ol 
tempeiature which is based entirely on theimoclynannc considera- 
tions, the condition dctei mining the steps being simply this, that 
the same amount of woik shall be done by the heat as iL pusses j 

down through each step. < 

i 

48. No change of Entropy in Adiabatic Processes It ibl- ^.(rf 
lows fiom the definition oi entiopy given m Ait 4.1 that uhcn 
a substance is expanded or compiessed in an adiabatic mannci 
(Ait 23) its entiopy docs not change An adiabalic hue is con- 
sequently a line of constant entiopy, 01, as it is sometimes called, 
an isentiopic line Just as isothermal lines can be distinguished 
by numbeis T a , T 2 etc denoting Lhe paiticulai tempeiatuie foi 
which each is chawn, so adiabatic lines can be distinguished by 
numbeis (j} l , 0, etc denoting the paiticulai value of the entiopy 
foi each 

We might accoidmgly define the enliopy of a substance as thai 
characteristic of the substance which does not change in adiabatic 
expansion 01 compiession, and this definition would be consistent 
with the method of leckonmg entiopy desciibed in Ait 44. Il is 
only in a icveisible piocess that the change of entiop\ of a sub- ^-fS? 

stance is to be deteimmed by lefeience to the heat it takes in 01 
gives out. The definition of an adiabatic process (Ait. 23) excludes 
any piocess that is not reveisible 

49 Change of Entropy m an Irreversible Operation It is 
impoitant in this connection to leahze that a substance may m- 
ciease its entiopy without having any heat communicated to it 
fiom outside When a substance expands in an it reversible manner, 
as by passing tlnough a thiottle- valve fiom a legion of high pies- 
suie to a region of lower pressuie, it gams entiopy Woik is then 
done by the substance on itself, in giving energy of motion to each 
portion as it passes through the valve, and this energy of motion v 

is futteied down into heat as the motion subsides thiough mteinal | 

fuction. The effect is like that produced by the communication I 



49 



some heat, though none is taken in from outside the substance. 
pansion through a, throttle- valve may be rcgaidcd as consisting 
two stages The fust stage is a moic 01 less achabatic expansion 
ling which the substance docs work in setting itsell in motion 
second stage is the loss of this motion and the consequent 
iciation within the substance itself of an equivalent amount 
heat Thcie is accoidmgly a gam of entropy, which occuis 
3anse the process as a whole is not IC T , cisiblc 
\VL cannot dncctly apply the definition of entiopy given in 
I L4 to deteiminc the amount by which the entropy of a sub- 
nee changes in an nieveisible opeiation such as thiotthng. 
t when the linal state is known it is m gencial easy to calculate 
1 cntiopy coiiespondmg to that state, by considering the amount 
which the cntiopy would ha\c changed if the substance had 
nc to thai state by a ie\eisible opeiation foi which $dQ]T 
amines the change 

,Vhcn a substance has passed thiough any complete cycle of 
lations Us entiopv is Hie same at the end as at Lhe beginning, 
the oiiginal state has been rcstoiecl in all ict>pects This is tine 
in mcveisibk cycle as \vell as of a ic\eisible c^cle But toi an 
\usible c^clc jdQ/T does not vanish It has a ncgatixe value 
I l'3) and il docs not mcasinc change of entiop} , foi it is onK in 
inlcmalh ie\eisible action that the change ot cnhopyis 



Sum of the Entropies in a System It is mstiuctne to 
how the sum ol Hie cntiopies ol all paits ol a theimo- 
i.inuc system is allected \\hcn we include not onh the \\oikmg 
slancc but also the soincc ol heat and the sink 01 recen ei 
which heat is i ejected CoiiMdei a evchc action m \\hich the 
king substance takes in a quantity of heat Qi from a souice 
l\ and i ejects a quantity Q 2 to a sink at T z When the cycle 
omj)lefcd the source has lost cutropj to the amount QJT l tlic 
king substance has ictumcd to the initial state, and theicfoie 

neithei gamed noi lost cntiopy the sink luib gamed entiopy 
lie amount <2 2 /T 2 If the cycle is a reveisiblc one Q i /T 1 = Q 2 /T 2 , 

thcicloic the system taken as a whole, consisting of souice, 
stance and sink, has suffeied no change in the sum ol the 
topics of its paits. But if the cycle is not icversible the action 
?ss efficient, Qa bears a l al ei piopoition to Qi ^nd Q 2 }T 2 is 
iter than Qx/^i Hence m an irieveisible action the sum of 
entiopies, of the system as a whole becomes increased This 



E T 



50 llllkKMUlJ*r\AlUlA,S [LH. 

conclusion has a veiy wide application it is Line of any system 
of bodies in which Lheimal actions may occui U may be 
expiessed m geneial teims by saying that when a system under- 
goes any change, the sum of the entiopies of the bodies which 
take pait in the action lemains unalteied if the action is reversible, 
but becomes mci eased i( the action is not icvcisible No leal 
aclion is stnctly icveisible, and hence any ical action occui- 
img within a system ol bodies has the effect of mci easing the 
sum of the entiopies of the bodies which make up the system 
This is a statement, in teims of entiopy, of the pnnciple that in all 
actual transfoimations ol cneigy there is what Loid Kelvin called 
a universal tendency towaids the dissipation of eneigy ' Any 
system, left to itself, tends to change in such a manner as to inciejse 
the aggiegatc entropy, which is calculated by summing up the 
entiopies of all the paits The sum of the entiopies m any system, 
consideied as a whole, tends towaids a maximum, which would be 
i cached if all the eneigy of the system wcie to take the fonn of 
umfoimly dilluscd heat, and if this state were i cached no fuithei 
transfoimations would be possible Any action within the system, 
by increasing the aggiegate entiopy, bungs the system a step 
nearer to this state, and to that extent diminishes the availability 
ol the eneigy m the s}stcm foi finthei liansloimations. 

This is tine of any limited system Applied to the umvcise 
as a whole, the docLnnc suggests that it is m the condition o( a 
clock once wound up and now miming down As Clausius, lo 
whom the name cntiopy is due, lias icmaikcd, '"the enemy of the 
univeise is constant the cntiopy of the- univcisc tends towmds 
a maximum." ^W 1 ', 3"* 

An extiemc case of thcimodvnamic waste oca us in Ihc dncct 
conduction of a quantity of heat Q liom a hot pai t of Ihc sj stem, 
at 2\, to a coldci pait at T,, no woik bcmq done in the pio- 
cess The hot pait loses cntiopy by Ihc amount Q/^\ the cold 
part gams entiopy by the amount Q/T 2 , and as the lallei is 
gicatci thcic is an incicasc in the ay^u-galc qiianlily ol cntiO])y 
in the system as a \\holc. 

51. Entropy-Temperature Diagrams. We shall now con- 

sidci, in a moie general mannei, diagiams in which the action of 

a substance is exhibited by showing the changes of its cntiopy in 

relation to its tempeiatnre Such a diagiam fonns an mteicstmg 

* AlalhemafiLCil and Physical Papeis, vol r, p 511 



[j FIRST PRINCIPLES 51 

md often useful alternative to the pressure-volume or indicator 
iiagram. One example, namely the entiopy-tempeiature diagram 
or a Carnot cycle, has aheacly been sketched in fig. 5. 

Let dcf> be the small change of entiopy which a substance under- 
;oes when it takes in the small quantity of heat dQ at any tem- 
>eiaturc T, it being assumed that in the process the substance 
indeigoes only a leversible change of state. Then, by the definition 
f entiopy (Ait U), dQ 

( '' < r = ~f ' 

thence Td(j) = dQ, 

nd JT<ty=Jd& 

'ie mtegiation being perfoimed between any assigned limits. 
low if a cuive be diawn with T and </> for coordinates, jTf/<ji is the 
ica undci the cuive. Tins b} the above equation is equal to Jf/Q 
Inch is the whole amount of heat taken in while the substance 
isses Lluough the states which that poition of the cuive repre- 

nts Let ab, fig. 7, be any poition of the cuiv e oi and T. The 

ca of the cioss-hatched ship, whose bieadth is S<^> and height T, 

T8(f>, which is equal to 8Q, the heal taken in dining the small 

lauge 8cj} The whole area niabii 01 

V/0 between the hmiLs a and b is !/ l 

e wliole heat taken in while the sub- 

mce changes in a icveisiblc mannci i 

3111 the state icpicscntcd b) a lo ; 

c state lepicsented by b Sinn- | 

[y, in changing icvcisibly horn 

lie b lo state a by the line ba the 

bslancc icjccts an amount of heat n, t \n _ 

i 1 1 1 7 ElltUl/HJ I 

uch is mcasLiicd by the aiea nbam 

. . , . ., Fie 1 Enliopy Temperature 

ic base line ox conesponds to the fa Cm\o 

soli lie 7Cio ol tcmpciatmc. 

^Vhcn an entiopy-tempeiatinc cur\c is diawn foi any complete 
j lc of changes it foims a closed figuie, since the substance icliuns 
its initial state. To find the area of the figuie we have to mte- 

tc thioughout the complete cycle, and piovided theie has been 

nre\ r cisiblc action Avithm the substance, 




being the heat taken in and Q 2 the heat rejected. But the 
eicncc between these is the heat conveited into work, hence 



42 



52 THERMODYNAMICS |cn. 

when the mtegiation extends round a complete cycle and W is 
expiessed in theimal units. Thus an entiopy-tempcratme diagram, 
so long as it lepresents changes of state all of which arc icvcrsiblc, 
but not otheiwise, has the impoitant piopeily in common with a 
piessuie^olume diagiam that bhe enclosed area measures Lhe woik 
done in a complete cycle 

But the entiopy-tempeiatuie diagiam has an adv.mlage no( 
possessed hy the piessuie-^olume diagiam, m that iL exhibits nol 
only the work done, but also the heat taken in and the heat 
i ejected, by means of aieas nuclei the cuives An illustiation of this 
has aheady been given in speaking of the Gamut cvcle (Ail <10), 
and others will be found in Chapter III. 

52. Perfect Engine using Regenerator Besides the cycle <>P 
Cainot theie is (theoieticalty) one othci way in which an engine, 
can woik between a souice and icecivci so ab to make Ihe whole 
action icveisible, and theiebv tiansform into woik the gicalest 
possible piopoition of the heat that is supplied Suppose their is, 
as part ol the engine, a body (called a tk icycneialoi ") into \\lnch 
the woikmg substance can tcmporanh deposit heal, uhile. I ho 
substance falls mtempeiatuie fiom the upper limit I\ to the lowci 
limit T-,, and suppose lurthei that this is done in such ,1 in. unit i 
that the tiansfer of heat fiom the substance to the icguicialoi is 
leseisible. This condition implies that thue is to be no sensible 
chffeicncein temperatuie between the woikmg subsUme and UK 
matenal of the regcneiatoi at any place wheie they aic in I luini.il 
contact. Then when we wish the substance to pass back horn 7', 
toT^we mayre\eise this tiansfcr, and so icem u Ihe lic.il which 
uas deposited m the icgeneiatoi This alteinate stoimo ,,iul 
restoring of heat scives instead of adiabalic expansion ami com- 
piession to make the tempciatme of the woikmo siihsl.mou pass 
fiom 2\ to T z and fiom T, to f l\ icspeetivcly It t nables UK- inii- 
peiatuie of the substance to lall to 2\ bcfoieheal is K |(ckd lo llu 
icceiver, and to use to 2\ bcfoic heat is taken in liom UK soincc 

This idea is clue to Robcit Stilling, A \h o m ia-27 (Usi 4 nt'il an 
engine to give it effect Foi the pieCent jnnjwse il will snlluc lo 
descube the legeneiatoi as a passage (such as a group ol lubes) 
thiough which the woikmg fluid can Iravcl in cither dm-chon 
whose walls have a vciy laige capacity foi heat, so that the amoiinl 
alteinately given to 01 taken from them by the woikmg fluid 
causes no moie than an insensible rise 01 fall in their tempcralme 



ij FIRST PRINCIPLES 53 

The lempciatuic of the walls at one end of the passage is T 1} and 
this falls continuously down to T z at the other end. When the 
working fluid at temperature 7\ enters the hot end and passes 
tlnoii>h, it comes out at the cold end at tcmpciatuvc T z , having 
stored in the walls of the regcncialoi a quantity of heat which it 
will pick up again when passing tlnough in the opposite dnection 
Dm mg tlve leluvn journey of I he working (hud through the ic- 
gcntraloi fioin (he cold to the hot end its temperatuic uses fiom 
T, to 7\ by pickino up the heal which was deposited when the 
woikino fluid passed through fiom the hot end to the cold The 
piocoss is shicllv rc\ cisible, or i at her would be so if the legcneiatoi 
had an unlimited capacity for heat, if no conduction of heat took 
place along Us walls fiom the hotter paits towaids the cold end, 
and it* Lhcre wcic no loss by conduction or radiation fiom its ex- 
ternal smfacc A icgcnciatoi satisfying these conditions is of 
conisc nn ideal impossible' to icalize in piacticc. 

53- Stirling's Regenerative Air-Engme. Using an as the 
woiking substance , .ind <mplo\ing his icgenuatoi, Slnling made 
<iu enonu which, allowing loi piaclical impel lections, is tlie 
eaihesl example of a icxeisible engine Tlie c\ cle of opciations in 
Milling's engine nas substantial^ this (in describing it ue tieat 
an as a pufecl gas) 

(1) An, \vlueh h;id luui heated to T } b\ r passing tlnouqh the 
K'i>riKM at 01, ^v t ls allowed to o pand isollu i mall} lluoiioli .1 ialio ;, 
lakiiiH in lu.il fiom a Imiuee and laism^ a pislon Ilcnt taken 
in (pel 11) <>l an) - R r L\ lon i / (b\ Ait 2<S) 

(2) I'lu' dii was causal to pass thiouyli the icgcneiatoi fiom 
Ilic hot to the cold end, deposit ing heat and having its tempeia- 
hiu' loweied to T, without change of volume. Ilcat stoicd in 
legc'iieiuloi -- A",, ( r l\ T,) The piessuie of comse fell in propoi- 
lion lo Uu Jail in Icmpei.iluic 

(,'3) The air was then compi essecl isothcrmallv at T z , tlnough the 
same ratio ? to ils onginal volume, m contact with a icceivcr of 
heal. Heat rejected = JtT. 2 log c / 

(L) The air was again passed through the legenerator fiom 
the colcl to the hot end, taking up heat and having its tcmpcratuie 
laiscd Lo r l\ Heat restored by the regenerator = K v (2\ T 2 ). 
This completed the cycle 



THERMODYNAMICS 



[fir. 



The efficiency is 



n j log t i - RT 2 log, r __ Tj. - T z 




Volume 

"Fifi 8 Ideal Hulii nl 01 (lia"iiiin 
of An EIU-UUI \villi Ili'ji-iMim- 
afioi (Sdiliuj 



is 



RT, log, 7 T! 

Themdicatoi diagram of this action is shown in ho. 8 

engine is impoitant, not as a piescnt- 

day heat-engine (though it has been 

icvived in small foims aftei a long 

mtenal of disuse), but because it is 

typical of the only mode, other than 

Camot's plan of achabatic expansion 

and adiabatic completion, bv which 

the action of a heat-engine can be made 

leveisiblc 

A modified foim of legeneiative en- 
gine was devised latei bv Encsson, who 
kept the piessure instead oi the volume 
constant while the woikmg substance 
passed through the legenciatoi, and so 
got an indicator diagiam made up ol 
frno isothennal lines and two lines of constant pussuu 
The entiopy-tempeiatiuc diagiam of a ie<>cuci;ili\ t 
of the type shown m fig. 9 

The isotheimalopeiationof taking in heal air, is irpusui ltd l n 
ab, be is the cooling of the substance from 
TI to T z in its passage thiough the ic- 
geneiatoi, \\heie it deposits heat cd is the 
isotheimal i ejection of htat at T Z) and da 
is the icstoiation of heat by the icgcncialor 
while the substance passes thiough it in 
the opposite dnection, by which the 1cm- 
peiatnie of the substance is laised fiom T, 
to T! Assuming the action of the ic- 
geneiatoi to be ideally pcifect, be and ad 
aie piecisely similai cuncs wh,itc\ei IK- 
then foim The aiea of Ihe fio mc is then 
equal to the aiea of the iccluns-le which 
would icpiesent the oiclmary Cainot cycle 
(fig- 5) The equal areas pbcq and ndam measmc Ihe lunl slou-tl 
and lestoied by the regenciatoi 

When the woikmg substance is an and the icffcncmlivc chaum-s 
take place eithei undei constant volume, as in Stiilmg's engine, 



/ 



(llll< (111 
UllL'Uli) i 

ialoi 



Fit <f< Hd 



FIRST PRINCIPLES 



55 



under constant piessuie, as in Encsson's, the specific heat K 
nng treated as constant, ad and be are logantlimic ciuves with 
ie equation 



</> = 



^ -f constant, 



being K v in Stirling's process and K$ in Encsson's. 

54 Joule's Air-Engine. A type of an-engme was pioposed 
r Joule which, ibi seveial icasons, possesses much theoietical 
teiest. Imagine a chambci C (fig 10) full of air (temperature T 2 ), 
Inch is kcpL cold by circulating watei or otherwise, anothei 
ambei A healed by a furnace and full of hot an in a state of 




Fi 10 Jouln's 



npicbsion (lunpuatuic r l\), a compicssing c\hndci M by \\lnch 
may be pumped fiom C into A, and a \\oikmg c^hndei N in 
leh an liom A may be allowed to expand bcfoic pasMiiir back 
o tlic cold ch.imbci C We tJiall suppose the chambeis A and 
o be laigc, in oompaiison ^vith the volume of an that passes in 
l\ ibliokc, so that the picssuic in each of them may be taken as 
isibly constant The pump M takes in an fiom C, compi esses 
adiabatically until its press me becomeb equal to tlie piessuie 
A., and then, the \alve v being opened, dchveis it into A The 
licator diagiam foi this action on tlie pait ol the pump is the 
giam/c/w? m fig 11. While this is going on, the same quantity 
hot an fiom A is admitted to the cylmdei N, the valve u is then 
sed, and tlie air is allowed to expand adiabatically in N until 
piessuie falls to the prcssuie in the cold chamber C, During 



56 THERMODYNAMICS [c'ir. 

the back stioke of N this an is disciplined into C The opcinlion 
of N is shown by the indicator cliayiJiin cbfj m h" 11. The nioa 
fdae measuies tlie woik spent in diivino I ho pump, Ilio aioa <// 
"is the woik done by the an in the woikino c-yhiuUi N The 
dil'feience, namely, the aiea tthid, is I ho nol amoiml of w.ik 
obtamedby canyinn the <>ivcn qnanlily of an Ihionidi ,-i eompkle 
cycle Heat is Lakcn in when the an has ils lempeial uio taisul 



a _Jt 




V 

Fig 11 Indicaloi dm<j;i am in Joiilo's An Kii'.'Mir 

on entenng the hoi thambci / Since llns happens al .1 pussnu 
which is bensibly conslanl, Ihe heal lakcn in 

whcieT B = T 1 , Ihe Lcmpeialme ol /, and '/' is llu l< nipi i.ihne 
i cached by adiabalic compulsion in Ihe pinup Similail\, Ihe 
heat lejcclccl Q - f (T T } 

whcie T d To, Lhc Icmperalme ol (', ami 7', is UK l< mpt lalnu 
i cached by admbalie expansion in N Sinei Ihe (\pansion ,uid 
compiession bolh lake place be I we en lliesami liiiiini.il pussiius, 
the lalio of expansion jmel ooinpiession is Ihes.une ( .dlini; il ?, 

we have 7' r j\ 

i, y-! 

T , T 

- 1 il * 

(Art 26), and hence also 

/Ti fit ni ri\ rii ri\ 

B o i It (i a ~ * it 

- >., , and 

- 1 a J il 'a J il 

Hence ' , /' _- r /' ; 

and the effieiency 




i] FIRST PRINCIPLES ^ 

This is less than the efficiency of a perfect engine working between 

/HP /7"T \ 

the same limits of temperature f l ~ 2 J because the heat is not 

taken in anel rejected at the extieme temperatuies 

The atmospheie may take the place of the chamber C that 

ib to bay, instead of having a cold chambci, with 

circulating watci to absoib the icjecteel heat, the 

engine may ehaw a ficsh supply at each stroke 

fiom the atmosphere, and chschaige into the 

atmospheie the air which has been expanded 

achabatically m N 

The entiopy-tempciature ehagiam foi this 

cycle ib chawn m fig 12, where the letters icfei 

to the same stages as m fig 11 After aehabatic 

ccmipicssion da, the air is heated m the hot 

chambci J, dnd the cuixe ab foi this piocess 

has the equation Fig 12 Enfciopy- 

toinpeiabiiici flia- 
/2 ' K pdT , . , > giam in Joulo's 

~7j\ = ^ Jl ("'gi ~~ '0 l a) All EngllU.' 

Then aehabatic expansion ones the line hi, diul cil is anolhel 
logaiilhmic cinvc Joi the i ejection ol heat to C' by cooling unclii 

T T , ('& 

constant picssiuc The latio ^ , which is icpicscntcd by ^ in 

fio 11 anel by "~ m fig 12, shows the propoibion which the 

J nb 

volume of the pump J/ must bcai to the xolumo of the \\oikmg 
cylmdei N. The need ol a huge pump \\oulcl be a si nous diax\- 
backiu [>i act ice, foi it would not only make the engine bulky bul 
would cause <i iclatixely Luge pait of the net indicated \\oik (o 
be expended m oxeicoming inction xvilhm the engine itsell 

In the- oiigmal conception of this engine by Joule it uas in- 
tended that the heat should icath the woikmg an lliibugh llu 
walls of the hot chambci, fiom an extcinal soiuce. Hul mslrad 
of this we may have combustion ol luel going on w.thin the hot 
chamber itself, the combustion being kept up by the supply of 
ficsh an which comes in through the compicssing pump, and, ol 
couisc, by supplying fuel cilhei in a solid foim fiom time to him- 
tlnough a hopper, 01 in a gaseous 01 liquid form. In olhci 
woids, the engine may opeiatc as an intunal-comhuttion engine' 
Iiilcrnol-combusLion engines, essentially of the Joule type, e-m- 
ploy.ng solid fuel have been used on a small scale, but by far (he 



58 THERMODYNAMICS fen r 

most impoitant development of the type is, to be found in tMu>iiK^ 
which T\ork by the explosion or binning of a mivhnc of .111 \\\ll\ 
combustible gas or the vapoin of n coiiibushblr liquid Tin 
thermodynamics of inteinal-combushon engines Mill lx nm- 
sidered in a later chajatei. 

We shall also see later (Chapter IV) thai i puclic.iMr n- 
Ingeratmg machine, using an for working suhsl.mci 1 , is ulil.iiiK d 
by making Joule's An -Engine \voik as a licaL-punip 



CHAPTER II 
PROPERTIES OF FLUIDS 

55. States of Aggregation. In the previous cbaptei the only 
substances whose pioperties \veie discussed were imaginary ones, 
namely pcifect gases We have now to tieat of leal substances, 
such as steam, cai borne acid, 01 ammonia, which serve as work- 
ing substances in heat-engines 01 lefngeiating machines, and to 
examine then action and propcities in the light of theimodynamic 
pimciplcs. 

Any such substance may exist in thicc states oi allegation, 
solid, liquid and gaseous We aic mainly concei ned \\ ith the liquid 
and gaseous states, in eithei of Mhich Lhe substance is spoken of 
as a fluid The woikmg (hud in an engine is often a nuxtuie of the 
same substance in the h\o states of liquid and \ apoui , but in some 
stages of the action it may consist cntnelv of liquid, in otheis 
onlncly of \apoui The vapoui of a substance may be either 
sahualed 01 supei heated A vapom mixed \\ith its liquid, and in 
cquihbiium with it, must be satuiatcd Any attempt to heat the 
nnxtuic Avould icsult in moie of the liquid tinning into satmated 
vapoui But when a vapoui has been icmo\cd fiom its liquid it 
may be heated to any extent, thcieby becoming supeiheated. 
Thus when steam is fonncd in a boilci it is necessanly satuiated 
when the bubbles leave the watei, but it may be supeiheated on 
ibs way to the engine by passing thiough hot pipes which cause its 
tempciatme to use above Lhal of the boilei 

Any of the so-called peimanent gases, such as hydrogen, 01 
oxygen or mtiogcn, is a supei heated vapour which can be i educed 
to the satuiatcd condition by gieatly lowering its tcmpeiatme 

At any one pressme the saturated vapour of a substance can 
have but one tempeiature the snpci heated vapour at the same 
pressuie may have any temperature higher than that 

In the change of state from solid to liquid, and again in the 



60 THERMODYNAMICS [cir. 

change from liquid to vapour, heat is taken in, though the substance 
does not rise in temperatuie while the change is going on. The 
heat so taken in was said in the phiaseology of old wiitcis to be- 
come latent, and the name Latent Heal is still applied lo it. Thus 
the heat taken in by unit mass of a substance in passing, wilhoul 
change of piessiue, fiom the solid to the liquid slalc is called I he 
latent heat of the liquid, and the heat taken in by uiul mass m 
passing, without change of picssure, fiom Ihc slate of liquid 
to that of vapoui is called the latent heo,t of Ihc \npoui The 
latent heat of water is SO thermal units, which means that 11111 1 
mass of ice takes in SO theimal units while it mclls, Ihc theiiual 
unit being one-hundredth pait of the quantity of heal icqiinul lo 
waim a unit mass of water from to 100 ccntigiadc 

The tempeiatuie at which ice melts is only vciy slightly affeclcd 
by the piessine (see Ail. 99), and the latent heal of walei is 
piacticallv the same at all ptessuies ordinal ily met with. 11 we 
assume the piessiue to be one atmosphere, ice melts at Hie tem- 
peratuie which is taken foi the lower fixed poml (0 C ) in giadua- 
ting a theimometcr (Ait 15) 

At a picssuie ot one atmospheie watei boils at Ihc Icmpcialme 
which is taken ior the uppci fixed point of the Ihc-Tinomelei 
(nameh 100 C ), and the latent heat of the vapoui is 539 y I hei mal 
units We shall see immediately that the tempeiatuie ;i( which UK 
change fiom liquid to vapoui occuis, and also the amount ol heal 
taken in dining the change, depends gieatly on the piessmc Al 
higher piessuies the tempeiatuie of boiling is lughei and the 
amount of latent heat is less 

In desciibing the piopeilies oi fluids it will save cncuniloeuhnu 
to speak usually of water, taking it as Upical ol the icsl II is 
itself of special mtciest to the cngmeci, being the \\oiking substance 
of the steam-engine, and the numciical 'tallies by winch its pro- 
peities are expicsscd aie bcttei known than those th.it u-lnle to 
other fluids But the definitions and thcimodyiunuc.il pimeiplcs 
which will be stated must be undcistood as applying lo fluids m 
geneial. 

We have now to considei m more detail some of Ihc points lhal 
ha\e been bnefly summaiizcd in this Aiticle 

56 Formation of Steam under Constant Pressure. The 
piopcitics of steam, or of any othei vapour, are most conveniently 
stated by lefeiimg m the fhst instance to what happens when il is 



Ij Jt'JLtU.l'JiJLlJLiJto Ul< JbL/UJLUb 51 

brined undo constant piessuie. This is substantially the piocess 
iv Inch occurs in the boiler of a steam-engine when the engine is at 
woik To fix the ideas we may suppose that the vessel in which 
> Learn is to be foimed is a long upiight cylinder fitted with a 
Cuctionless piston which may be loaded so that it exerts a constant 
prcssme on the fluid below Let theie be, to begin with, at the foot 
of the cylmdei a quantity of water (which foi convenience of state- 
ment we shall take as one unit of mass, 1 lb. say), and let the piston 
rest on the surface of the \\atei with a piessuie P. Let heat now 
be applied to the bottom of the cylmdei As heat enteis the water 
it produces the following effects in three stages- 

(1) The temperature of the watei uses until a ccitam tem- 
peiature I\ is leached, at which steam begins to be foimed The 
value of r l\ depends on the pailicular piesbine P which the piston 
excits Until the tcmpeialmc T^ is i cached there is nothing but 
Avater below the piston 

(2) Steam is loimed, moie heat being taken in The piston, 
which is supposed to continue to e\eit the same constant piessuie, 
uses No fuithcr incicasc of tempeiatuic occuis dining this stage 
which continues until all the watci is coin cited into steam Dining 
this sUigc the slcam which is foimed is saluiatecl The \olume 
which the piston encloses at the end of this stage the \olume, 
namely, of unit mass of satmated steam at piessuie P and con- 
sequently at tcmpciatmc r l\ will be denoted by V 6 

(3) If moie heat be allowed to cnki aftci all the water has 
been coin cited into steam, the \olumc will incicasc and the tcm- 
pciatuic will use The slcam is then <,upei heated its tcmpciatme 
is above the Icmpciatmc of satiualion. 

57. Saturated and Superheated Steam The difleience 
between satuialcel and supei heated steam may be c\picss>ed by 
saying that it walci (at the tcmpeiature oi the steam) be mixed 
with slcam, some of the watei will be cvapoiated if the steam is 
supciheatcel, but none if the steam is satuiatccl. Steam in contact 
with watci, and in thermal equilibimm Avith it, is necessarily 
satuiateel When satuiatccl its properties differ considerably, as 
a rule, from those of a peifcct gas, but when supei heated they 
appioach those of a peifcct gas moie and moie closely the faithei 
the piocess of superheating is earned, that is to say, the more the 
tempeiature is raised aboveT s ,the tempeiatuie of satiuation corre- 
sponding to the given pressuie P. 



02 11-iJbJttlMUJJYlMAiVllLb [C'H. 

58. Relation of Pressure to Temperature in Saturated 
Steam. The tempeiature T s at uhich steam is 1'oimcd under the 
conditions described in Ait. 56, which is called the tcmpeuiLiuc of 
saturation, depends on the \alue oi P. The i elation of pi ethnic to 
the tempeiatine of satuiation ^as deLcimincd \\ith great caie by 
Regnault, in a senes of classical expeiiinenls Lo which much of oui 
knowledge ot the pioperties of steam is due" 1 . Rcgnault's obsci- 
vations extended horn tempera tuies below the zeio oi the ccnti- 
giade scale, \Uieie the vapour whose piessme was measuicd was 
that given off by ice, up to 220 C. The pressmcs found by him, 
expiessed in millimeties of meicuiy, weie as follows, omitting those 
below C. as not iclcvant to steani-cngmc calculaLions . 

Pietwiio of animated sU'tini 
Tempeiatiue C m mm ot Moiomy 

460 

25 23 55 

40 54 91 

50 91 98 

75 288 50 

100 700 00 

130 2030 

160 4051 

190 9420 

220 17390 

It will be seen from these iiguics that the picssinc of satuialul 
steam ns.es with the tempeiatine at a late which increases rapidly 
in the uppei regions oi the scale. Vanous empnical loimulas have 
been deviled to expiess the i elation ol pi ess, me Lo tcmpcuitme m 
satmatcd steam and to allow tables Lo be calculated in which inLci- 
mediate values aie shown When a table is available, houcvci, it 
it, moic comement to find Lhe piessme coiicspoiidino Lo n ivcn 
tempeidtme, 01 Lhe Lcmpciature coiicsjiondmg to a n iv un JHCSSIMC 
dnectly fiom it, eithei mtcipolaLing or chawing a poition oi Liu- 
cuive connectmo picssuie with tcmpciatmc when I he \ nines con- 
ceined lie between those that are stated m Lhe table. 

59. Tables of the Properties of Steam. At the end of this 
book a numbei of Tables will be iound showing not only the ic- 
lation of the piessme to the temperature of saturation, but also 
various other pioperties oi steam which aie of use in enginccimg 

* Him. Imt F,a>ice, 1847, l XS1 An accouafc of KognauK/6 mothoda of 
espenment and a statement of lus results expressed m Bntish moiiamou will Lo 
found in Dixon's Tieatise on Neat (Dublin, 1S49). 



n] PROPERTIES OF FLUIDS 68 

calculations. Tables of the propeiLies of steam have been calculated 
by Piofessoi Callendai, by methods which will be explained later, 
and have been published undei the title of The Callendai Steam 
Tables '. Fiom Callendai 's babies, which give the most authon- 
tative icsults now available, a selection has been made, with his 
pei mission, foi the piuposes of this book. 

The fig m cs which are given foi the piessure of saturated steam 
at vauoub temperatincs aie not taken directly fiom the measure- 
ments of Regnault, but aic infeired fiom a charactenstic equation 
which Callcndar has devised to expiess the i elation between pres- 
suic, volume and tempeiatuic within the woikmg lange The 
validity of that equation (within the range to which the tables 
apply) is demonstiated by the general agieement of the quantities 
calculated horn it with the best expcnmental icsults, in measure- 
ments not only of the pressuie at saturation but of other properties 
ol steam. The picssuics, howevei, which aie stated in these tables 
do agree \civ closely with the icsults of Regnault's obseivations 
quoted above It is only at the highest piessures that an appieci- 
ablc difference will be lound, and c\'en thcie it is not matenal 

In othci icspccls the Callendai tables will be found to diffei 
somewhat widely fiom the eailiei tables of such authoiities as 
Rankme | 01 Zcuner |, which have been accepted as standaids and 
copied into many text-books When these were calculated the only 
a\ ailable data of value wcie those kunished by the expenments of 
Reimault But moie iccent icsearches have supplied additional 
data which in some paiticulais modify his, and it is now clcai that 
Regnault's hgtiics lequue icusion and m some cases consideiable 
anicndnicnl The vanous piopcities of ste.un, 01 of any othei 
vapom, aie linked together in such a mannei that the lelations 
bctucui lliein must satisfy ccilam theimocl^ namic equations This 
alfoids a tcsl of consistency, and in the light of such investigations 
the (iguies given in the old tables aie now known to be not even 
mutually consistent Callendai 's tables give a set of values that 
aic consistent amongst themselves and aie also m good agree- 
ment Avilli I he most tiustwoilhy cxpeiimenial lesults Further ic- 
scaiches may in Lime lead to a still closei adjustment of the figuics 
to the u'sulls of observation, but Callendai's \alucs lor the various 

* London, Edward Arnold, 1916 Students should obtain a copy of those 
Tables, which contain fulloi pailiculara than arc quoted hpre 

f- Rankiue, A Hatmal of the Steam Engine and other Prime Movers 

f Zouner, Techmsohc ThermodynamiL, vol n (Tians by J F. Klein, 1907 ) 



[CII. 

quantities may be accepted not only as mutualty consistent, fiom 
the Lliermoclynamic point of view, but as ceitamly collect enough 
for the purposes of the engmcei. 

60. Relation of Pressure to Volume in Saturated Steam. 
Among the quantities shown in the tablet, is the volume F 4 , in cubic 
feet per lb., of saturated steam at vaiious temper atuies and at 
vaiious pressuies The volume of a given quantily ot saturated 
steam at any assigned temperatme or pressure is a quantity 
difficult to measiue by diiect expeiimcnL, and the volumes which 
aie given m steam tables aie generally mfcned from the results of 
expenments on other piopeities which can be moie easily measured. 
Successful mcasuiemcnts of \ olume ha\ eliowe~\ er been caincd out ' 
and the icsults aie in geneial agieement with the figures stated in 
these tables. 

The iclation of P to V s in sat ma ted steam is appioximately 
expiessed by an empnical formula 

PF s lu = constant. 
With P in pounds pei sq inch and V a in cubic feet pci lb this gives 

PV a l * =490. 

WiLhP in kilogiammcs pei squaicccntimetie and V s in cubic meties 
pci kilogiammc, it becomes 



This ibiniula applies, well liom a piessuie of say 1 pound pci 
squaie inch up to SOU pounds pei scmaic inch Within these Jimils 
it gi\ es "\alues which agiec to one pait in a thousand with those I 

in the tables. ^^ 

The student will find it useful to chaw cuives, with the data ? , 

of the tables, showing the i elation bet\vcen the picssure and the J 

temperature oi sat mated steam, anel also the iclation of piessuie to ,' 

volume, especially within the lange usual in steam-engine piacticc 
lie will obsen e that the late of change of piessuie with icspect to , 

change ol tempeiatuie mcicases lapielly as the tcmpeiatmc uses, * 

and lience that in the uppci pait of the range a veiy small ele\ ation 
of temporal ure in a boilei is necessnnty associated with ti large 
mciement of press me 

The piessuie shown by a piessure-gauge on a boiler is the excess ',* 

of piessuie in the boilei above the press me of the atmosphcic. 

* See especially Knoblauch, R Lmdo and H Klebe, lUittedungtn ilbcr For- 
sUuinijsarbeiteii hcraitsgeyeben vom V&em dadsclicr Ingemcurc, Heft 21, 1905 



r] PROPERTIES OF FLUIDS 65 

xonsequently the true or "absolute" piessure in the boiler is to 
>e found by adding, to the leading of a correct gauge, the piessure 
/Inch coriesponds to the height of the barometer at the time, this 
3 generally about id- 7 pounds per square inch or 1-033 kilogrammes 
'6r square centunetie. 

61. Boiling and Evaporation. The familiar case of water 
oiling in a kettle 01 other open vessel is only a special example of 
lie foimationof steam under constant pressuie. There the constant 
icssuie is that of the atmospheie, and consequently the temper a- 
iire at which the water boils is about 100 C ^ 

Watei in the open evapot cites slowl}*" at any tempeiature lower 
lian that at which it boils. Though the presume of the vapour so 
nmed is lowei than that of the atmospheie and may be veiy 
inch lower the vapour is able to escape from the suiface by 
if fusion the atmospheie is not displaced and the piessuie on the 
irface of the water is still that of the an As the tempeiature of 
r atei in the open is laised this slow evapoiation fiom the suiface 
ccomes moie rapid, but it is only when the tempeiatuie teaches 
ic valucwhich coiicsj)onds(loi satuiated steam) to the gi\ en atmo- 
)licnc pressure Lhat the watei boils the \apom is then foimecl in 
ubbles at the piessuie of the atmospheie, and it escapes not by 
iffusion but by displacing the supei incumbent an 

62. Mixture of Vapour with other Gases: Dalton's Prm- 

iple In what has been said about the iclation oi picsbiue 

id volume to tempeiatuie in the satmaled stale, it has been 

ruined Lhat in the pioccss ol formation Lheie is simplv a mixture 

1 the liquid with its vapour, no olhci substance being present. 

his is substantially tuio in a steam boilei 01 in the e\apoiator of 

icfiigciating machine. But the case is diffeient when the A apoui 

is to mix with anothei gas or gases. A pnnciple discovered by 

altou Lhen applies, that the piessuie in any closed space con- 

iininq a mixture of two or moie gases at any given tempeiatuie 

vciy approximately equal to tlie sum ol the prcssines which each 

'the gases would exert separately if the others weie absent, that 

to say if each of the gases (at the same temperatuic) alone 

cupicd the whole space These pressure^, which aie added 

igcther to make up the actual pressure, aie called "partial 

* Water in the open boils at 100 C when the atmospheno pleasure has its 
mdard value, which coriesponds to a barometer reading (collected to 0C ) of 
mm at sea level in latitude 45, or 759 G mm in London (see Art 12) 

E. T. ^ 



06 THERMODYNAMlLb t i.i, 

piessmes." An impoitant instance of the application of Dalton's 
pimciple is consideied in the next article. 

63. Evaporation into a space containing Air* Saturation 
of the Atmosphere with Water- Vapour When watei cv.ipoi- 
ates in a closed space containing an, the piocess qocs on unlil ;i 
definite amount of it has become mixed, as \apoui willi Hie an 
aheady theie When this has happened and a state of e<|iiilil)iumi 
is leached, the an is said to be satuiatul Mith walci -\apom. The 
amount of watei-vapoin that a t>neii volume of <ni will lake up in 
tins way depend^ upon the tenipciatinc it is \eiv ncaih UK- s.inu 
amount as would be lequiied to fill the same space wilh salmalc d 
steam at that tempeiatme il the an weie not puseut J?y ])alloii's 
pimciple the piessme of the mixed oases, namely the an and UK 
watei-vapom mixed with the an, is ven> ncailv the same as UK 
sum of the piessaiies which each would excit sepaiatcly LluL is 
to say the piessuie in the given space aftei the wiLci-\ tiponi has 
been foimed is gieatei than the pies>suic whuh Hie an would c\eil 
in that space, if the watei-vapoui wcie not thcie, by an amouiil 
which is neailv equal to the piessuie of satinated sleam al Ihe 
tempeiatme of the mixtuie It is appioximatelv hue lo say Ihal 
each of the constituents of the mixed atmospheie in UK closed 
space behaves as if it occupied the \\holevolume, and conlubulcs 
to the piebsuie just as, if the othei constituent weie ahsenl This is 
veiyneaily accurate at 01 dmaiypicssuics It becomes Jess aec uiali 
vxhen thepiehsuie is high the amount of watei^apom zeqmud lo 
satuiate the atmosphcie is then somewhat less than Ihe iiilr uonld 
lequne. 

As an example, suppose an at 25 C (77 Fah ) lo be sal male d 
with uatei -vapour At that tempeiatme one Ib of salmaled sleam 
would (by the Tables) occupy 092-4 cubic feet, and lluieloiv O.K 
cubic foot weighs 0-001 JJ, Ib Consequently each cubic fool ( I he 
an takes up 000114 Ib of watci-vapoin m icachmj. UK sl.iU- 
of satmation at that tempeiatme And since the eoucspomlm.. 
piessuie of waters apoui is 46 pound pei sq inch, Ihr pn ssiur 
in an enclosed space containing this moisi an is qic.>lci by K, 
pound per sq. inch than it would be if Uie walei-vapo.u wcu- 
removed and the diy an alone weie left lo fill the same spae ( al 
the same tempeiatme In othei wouls 16 pound per sq meh is 
the "paitial piessme" of the watei-vapom piesent in Lhe an undei 
the assumed conditions. 



When the amount of watei- vapour present in air is less than 
tough to cause safciuation the watei -vapour is held in a supei- 
;ated state. If the tcmpeiature of the mixtuie be lowered, a point 
reached at which the air becomes saluiated, and any further 
wcimg of the tempeiatuie causes some of the vapour to be de- 
bited as liquid on the Avails of the containing vessel, 01 on any 
u Licles of dust that may be piesent. Any solid paiticles will sei ve 
nuclei for condensation. The water condensed on such nuclei 
ims a mist of minute diops which fall so slowly that they seem to 
i held in suspension The tempeiatuie at which watei begins to 
1 deposited fiom moist an is called the dew-point Condensation 
' some of the watei contained in an will also occur on any cold 
irface (colder than the dew-point) uith which the air comes in 
mtact this icsults fiom local cooling of the an close to the suiface 
question. Thus in a refngeiatmg plant with pipes that convey a 
-imd coldci than the fleering point thiough the waim atmospheie 
' the engme-ioom, a coating of ice forms lound the pipes Foi the 
me icason an effective \\ay to diy an is to make it cold and chain 
\ ay the watei condensed in the piocess, at the lowest tempeiatuie 
ic an icmains satmated, but the amount of watei lequned to 
tmatc it at a low tempeiatuie is vcivsmall,anrl uhen it is allowed 
become waim again without taking up moie water it will be fai 
um sal mation 

64 Heat required for the Formation of Steam under 
Dnstant Pressure 1 Heat of the Liquid and Latent Heat 
ctuin now to the imagmaiy expenmeni of Ait 56, \\heie steam 
fonncd under the constant piessure of a loaded piston, nothing 
it watei 01 watei-vapom being piesent and enqiine what 
nount of heat has to be supplied in each stage of the opciation 
i i\\cfif,t stage the substance is wholly in the condition of A\atei 
Inch is being heated fiom the initial tempeiature to T 6 , the 
mpcialuic at uhich Ihc second stage begins. Dining this fiist 
age the heat taken in (per Ib of the watei) is appioximatcly equal 

one thcimal unit foi each dcgiee by which the tempeiature of the 
aler uses It would be exactly equal to that if the specific .heat 

watei wci e constant and equal to unity, but this is not the case, 
t about 30 C the specific heat of water is less than unit}', it 
isscs a minimum value thereabouts of 9967, and then increases, 
^coming appreciably greater than unity at such temperatmes as 
e found in steam boileis Thus for instance to heat 1 Ib of water 



68 1 lUiltiVLUJJ X IN AUI A^- 



I 

from C. to 80 C. icqimes 79 9 theimal units instead of 80 On 
the othei hand, to heat it fiom C. to 200 C , undei 1 a piessuic 
sufficient to pi event steam fiom foiming, requues ncaily 20.3 2 
thermal units instead of 200. These figuies will indicate how 1'tu 
it is legitimate to estimate the heat taken in dining llic iirst sliiL>c 
as one unit pei clcgiee Moie accuiatc values oi the heal ol Ihc- 
liquid, that is to sav the heat taken in dining the fiist stn^e', can i 

be found by means of the Steam Tables (see Ait. 09). 

Dining this first stage, \vhile the substance is still liquid, ncai ly .ill 
the heat that is taken in goes to meicase the stock of uitei nal em i gy . 
Theie is scaicely any exteinal \\oik done, foi the \olume is only 
slightly inci eased. Thus m heating watei horn C to 200" ('. i 

(undei apiessuie of 225 2-1 pounds pci s<] inch) the volume ol I he- 
water changes horn 0160 cubic ft. pel Ib to 0185. The e\le-i nal *y* 
woik done dining this, heating is thcieioie 225 21 x 111, x 0025 
or SI foot-pounds This is equivalent to baiely 00 theimal mill, 
and is negligible in companion wilh the 203 2 umls of heal llul 
aic taken m j 
In the second stage, the liquid changes mLo satmale-el sle.im wilh- I 
out change of tempeialuic The heat lhat is laken in elm ing ih ls 
stage constitutes what is called the Latent Ifcat ol [lie- \.ipom 
We shall denote it bv L Values of the latent jic.it of ssihnalnl 
steam are gi\cn in the tables Foi steam foimccl umln ,1 pie-ssim- 
of one atmospheie (salutation tempciatiiic 100 C ) Ihelalc-nl lu.il 
is 539 6 withlo\\ei piessuies of formation il is gievilci, jud \\ilh 
lughei piessmes it is less At the end ol the second sl,ii>c Ibi sub- 
stance contains no liquid, it is spoken of as diy s.iluiale-d sleam 
at any eailiei point, \\ hen the substance consists p,n lly ol s.il male d 
steam and paitly ol watei, it may be spoken of as we I sk-ain 

The latent heat of a \apom may be tledix-d as I he amounl ol 
heat which is taken in bv unit mass ol Ihe liquid while il .ill e li.ui'x -, 
into satuiated \apour uncle i constant piessine, Ihe liquid b.mun 
been pieviouslv heated up to the tempi-mime al Jmli flu \.ipoin 
is fonned 

A consideiable pail of the heat taken m dm ing M,is pmerss , s 
spent in doing external woik, since Ihe substance expands ng.uusl 
the constant piessme P. It is only the icmamder ol the so-calh-d 
latent heat L that can be said to remain m the ilmd and (<> con- 
stitute an addition to its stock of internal cncigv The amounl f 
spent m doing exteinal work during the second stage- is \ 
AP (V V \ 

- rj - i \' i r ; Jj 



n] PROPERTIES OF FLUIDS 69 

whcie V s is Lhe volume of the saturated vapour and V w is the 
volume of the liquid at the same temperature and piessme, A being 
the factoi for coiu citing units of work into theimal units. The 
excess of L above tins quantity measmes the amount by which 
the internal eneigy inci eases dming the second stage 

Thus foi instance when water at 200 C and a pressme of 225 24 
pounds pei sq inch is converted into steam, of the 467 41 theimal 
units taken in, 47 61 units aie spent in doing external woik "" and 
419 8 units go to mciease the stock of internal energy. 

65. Total External Work done. In the two stages together 
the whole amount of exteinal woik done is to be found by taking 
the whole increase of volume and multiplying it by the piessure. 
If we assume that the watci is onginally at C. its volume may 
be taken as 01CO In converting water fiom C to saturated 
steam at 200 C. undei constant piessme the exteinal woik done is 
found thus to be equivalent to 47 67 theimal units this is 06 
units more than the external woik of the second stage, foi it in- 
cludes the small amount ahcad\ icfmcd to as ha\mg been done 
dming the fust slugc The nholc mciease of mtcinnl encigj , fiom 
water at C lo satinatcd steam at an\ tcmpcialiiic, is equal to 
the \\l\o\c amount of heat tal en in, less the eqimalent of the 
cxlunal \\oik done This in fact is onh a p.uticulai example of the 
gcncial pnnciplc stated in Ait 9, that \\hcn any substance ex- 
pands in any mannei. taking in heat anddoin" \\oik, the heat taken 
in is equal to the \\oik clone phis the mciease of internal cncigy 
In Ihc case hc'ie considcicd the aclion is oom<i on nuclei constant 
picssuio, but the statement applies to any change of state whatevei. 

66 Internal Energy of a Fluid No mattci what changes a 
substance may undcigo, its inteinal eneigy \\ill ictuin to the same 
\aluc when the substance ictmns to the same condition in all 
u'specls In othci woids the inteinal cncigy is a function of the 
actual slate of the substance and is independent of the way in 
which that slalc has been icachcd Thus the inteinal eneigy of 
1 11) of sal ura led steam at a paiticulai piessme is a definite 
quantity which is the some whcthci the steam has been foimed by 
boiling under constant piessme or in any othei mannei. Steam 
foimed in a closed vessel of constant volume, foi example, would 
have the same internal energy as steam at the same piessure but 
formed nuclei conditions of constant piessme, though the amountof 

i * The volume of the watei is 0185 cubic ft and of the steam 2 0738 cubic ft 
The value of AP ( F, - F, t ,) is therefore 47 61 



70 THERMODYNAMICS | t -ji. 

heat taken in dm ing its fomiation would be diffeicnl, fornoexlriual 
work is done in the piocess of foimation in a closed \rsscl <>l con- 
stant volume In that case the hc.it Liken in would hi equal to 
the increase of internal eneigy 

We have no means of measuiing the total stock of inlciual 
eneigy in a substance, and can deal only with chants m llu- stock 
But by taking some arbitiaiy staiting poml as ,i '/cio fioin \\lnch 
the internal eneigy E is icckoncd we can r>i\e/i'a nnineiical \ alue 
foi any other state of the substance. That value ically cxpicsscs 
the diffeience fiom the internal eneigy m the zcio slate The usual 
convention is to wnte E = when the substance is in llu liquid 
condition at a tempeiatuie of C, and al apicssmc equal lo I he 
vapoui-piessure coirespon cling to that tempcialiue \Ve may call 
this, foi bievity, the zcio state of the substance 

Following this convention we take E = toi waloi al C The 
\alue of E for satmated watci-vapom at C will llu n be 5<> |, 'Jl 
thermal units (see Tablet, in Appendix) Thai this apices \\ il h <>[ lu i 
figuies in the tables will be seen by consiclcuri" I he couvcrsuin ol 
watei at C to steam alfprC. uiulci constaul pussinc Tbc <ml\ 
heattaken in is L, which is 59 J 27 units, and ol tins I hcc \l CM n,i I woi k 
AP(F b - V K ) icpiesenU 30 06 units- the dilfciencr niusiiirs K 

Values of E foi saturated steam at vnnoiis li-nijjtialiiirs aie 
given 111 the tables It will be seen lhat they mcirasc slow^ \\-ilh 
the tempeiatuie. 



67 The "Total Heat " of a Fluid We co.nc now ID i 

function of the state of any substance, a fuiu.-l.on which is ol vn v 
gieat use in theimodynamic calculations. II Js frc-iH-i.illy callc d Ihc 
" Total Heat" and is repicscnled ' by I he IctkT/ 

The "total heat" / is defined lor any slate of (he subslancr ), v 
the equation = 



That is to say / is equal to the sum of [he intern,,] em^y ,,,, ( | || 1( 
exteinal woik which would be done if ll lc subsL.ncr could !,< 
imagined to staiL fiom no N olume al all and lo exp.uul lo Us acl,r,l 
volume under a constant pressuu- equal lo Us aclual 



t so 

the actual state, I is also a funclion of llu aclual sl,,|, Us value , s 
ndependent of how the state has been ,,,! r,, slnun <u 
exam^e, the heat taken m during fouuahon depends on how I he 
* Callendar , h, s Table, u a os // to iopioBO.il HUH funohon J u WI1W tlf Ul(1 Jilofc 



rl PROPERTIES OF FLUIDS 71 

.team is formed, but the "total heat" / depends only on the final 
'ondition The total heat can be calculated for any condition of a 
.ubbtance, whether in the slate of liquid or of saturated 01 super- 
icatcd vapoui. It is measiued in thermal units per Ib. Values of 
he total heat of saturated steam and also of water under satma- 
1011 pressure at vanous tempeiatmes aie given in the tables. The 
olal heat of steam increases piogicssively with the temperature, 
cither moie lapidly than does the internal energy. 

It follows fiom the definition of I that in the zeio state of any 
.ubstancc, at which E is reckoned to be zeio, / is not equal to zero 
jut to a small positive quantity depending on the volume of the 
iquid and its piessme at that state Since E is then zeio I is equal 

AP V , whcicP is the picssure at the zeio state, namely the 
/apoiu-prcssuie at C , and F is the volume of the liquid at C. 
ind piessme P Foi water this quantity AP Q V is quite negligible, 
amounting as it docs to 000146 theimal unit For caibonic 
icid it is about 1 theimal unit, foi ammonia and sulpliiuous acid 
t is much less 

68 Change of the Total Heat during Heating under Con- 
stant Pressure An mipoitant propcrt) ot the function I is that 
when an) substance is heated undei constant piessme the change 
jf / is equal to the amount of heat taken in To piove this, let Q 
be the amount of heat taken in while the substance expands undei 
joustant piessme P fiom a state in which the volume is F L and the 
inleinal eucigy is E-^ to anothei state in which the volume is V 2 
and the intcinal cnciy) is E 2 Then the amount of external woik 
done is P (V VJ and, by the conscivation of eneigy, 

Q = E,-E i + AP(V.-V^ 
which may be wnttcn 

Q = Ei + APV, - (E : + APVJ 

01 Q. = I? ~ AJ 

where Z^is the total heat in the fust state and / 2 is the total heat 

m the second state 

69. Application to Steam formed under Constant Pressure, 
from Water at C. The above pioposition applies to cveiy stage 
of the imaginary expcnmcnt of Ait. 50 Refemng to that expeii- 
mcut, assume that lo begin with there is undei the piston 1 Ib of 
water dt C. and at the pressure P at which steam is to be formed 
By definition of the Lotal heat, 

I = E + APV, 



72 THERMODYNAMICS [cli. 

E at the beginning may be taken as zeio" 1 . Hence the value of J 
foi the water at C maybe taken as APF Q . whcie F is the volume 
of 1 Ib of watei at C and P is the piessuie at which steam is lo 
be foimed. At the end of the fust stage 



\vheie I w lepiesents the value of/ toi water at the tcmpcinluie at 
which steam is about to form [ . When values of /, aie known I his 
allows Q 15 the heat taken m dining the fust stage, to bi inoic 
accmately calculated than b}*- the lough method of A) I G-i Values 
of I w are included in the steam tables 

Dming the second stage an amount of heat equal io L is la km 
in at constant piessuie, and the total heat changes fiom J w to I a , 
where I s is the total heat of saturated steam. Hence 

/. = L + I w 

= L + Q^ + APF 

The sum L + Qj is the whole heat of formation, in I he experi- 
ment of Ait. 56. Thus the "total heat" of steam is equal lo 
the heat of foimation uudei constant piessuie, plit^ ,i small 
quantity \\hichis the theimal equivalent of the woik Uuil would |H> 
done in lifting the piston fai enough to admit the onginal \oluiiu' 
of the watei The quantity APV Q foims a \uy small pail of UK 
' total heat" it is only 37 theimal unit when the tempciahue of 
foimation is 200 C and it is much less at lowei tern pei.i Lines 

These lemaiks and the following tabula i scheme \\\\\ st'i\t lo 
show how the total heat of satmatcd steam (01 other vapom) is 
i elated to the heat of formation under constant piessme Bui Hie 
student should accustom himself to think of the lolal heal wilhoul 
lefeience to any piocess of foimation, as a piopcrly which a 
substance possesses in its actual state n piopeily \\luc-h is pisl .is 
simply a function of the state as is the tempt ial me, 01 (he pic ssinc, 
01 the \olume, 01 the internal cncigv, or the cnlmpy, which we 
shall ha\e to considei piesentlvj. 

* The convention of Ait 60 makes 0=0 foi wiloi alOC and piosNino J>, lie,,, 
andpreasuioP.whioInslughoi 



1( , H si,,( 

does not cause the internal energy of wntoi atOC to dilku upptwiahly fiom /.mo 



In C'allendar's Tables tins quantity / JH win ten 7; 
T The function here caUed tbo total hoat /, namely K , JPK, W1H mlllfl( | llc . (M i ,, y 
Willard Gibbs (T,a, ta oj Ike Connecticut Academy, vol nr, Collated Xnu^tu 
Papers, rol i, page 92), and was fii st called Iho " Total 1 loat by C.llni.lat ( 
May 19ui vol v, p 50) Its gtoal iniporLanco in tcchmcal tliOL.iiodynain.CH 
emphas^d by Molber, who omplo 3 ed it m chaits foi ovl,,bilmg tho pu^u 
of steam and other .ubstanccs Tho uso of such ohaita will bo dwonbocl lutoi 



I] 



PROPERTIES OF FLUIDS 

Total Heat, / 



E 



73 



APV 



ternal eneigy 
inucd by the 
toi in being 
id fiom C , 



Intci rial encigy 

acqimed during 

change of state from 

watei to steam, 

L-AP(V a -V w ) 



External woik 

done during change 

of state from 

water to steam, 



External work 

clone while the 

watei is being 

heated, 



APV. 




Heat taken in during 
second stage, L 



Heat taken in during first stage, Q 1 

70 Total Heat of a mixture of Liquid and its Saturated 
/apour It follows from Ait 68 that while a liquid is being con- 
r ci I eel into \ apour, nuclei constant pi assure, the total heat 7 inci eases 
11 piopoihon to the amount of vapoui that is t'oimed At am 
nleinieduilc stage in the piocess, if we call q the fi action that is 
apou/cd and 1 - q the haction that is still liquid, the total heat 

>f Ihc niixtine is T 

ql s + (1 -(/)/, 

vhieh may be uiillcn , 

!, + & 

Similaih, Mhilc a \apoui is bemo condensed under constant 
Hcssmc, / becomes less by an amount mcasmecl by the heat 
i\ci\ out, \\lnch is ])ioj)oitional, at am mtei mediate stage, to 
he fi action then condensed 

71 Total Heat of Superheated Vapour When steam, 01 
ny othci \apoui, becomes superheated (as in the thud stage of the 
\peiiincnL of Ail 50) b^ continuing the heating piocess undei 
onslanl piessuic after 1he satuialcd condition has been leached, 
lie value of I becomes inci cased above the "<; alue 1^, by an amount 
qua! to the heat so taken in We might find the total heat of 
upei heal eel steam by calculating the supplemental y amount taken 
n eluung I he piocess of supeiheatmg, piovided we knew the specific 
teat of the vapoui dining the process of heating it, undci constant 
nessiiie, fiom its Lempeialuie of satin a I ion to its actual tempera- 
me Bui this specific heat is not a constant it diminishes slowly 
s the temperatuie uses, and it is greater at high pressures than at 
ow pressuics A better way of finding the total heat in supeiheateel 
team is to use an equation, devised by Callendar, which gives the 



74 THERMODYNAMICS [vu. 

total heat of the supei heated vapour directly for any condilion ol 
temperatuie and presume, without iclcicnce Lo the mode ol lonna- 
tion Tins will be descubed in a later cliaptci, and a selection of 
numeiical values will be found in the tables luom Ihcin I he heal 
taken in dining supeiheating at constant piessuie may be Jound 
as I' !, \\heie /' is the total heat m the supcihcatcd slale 
and I s the total heat in the satmated state at the same piessuie. 

In engmeeimg piactice, the supeiheating ol steam is generally 
earned out at constant piessiuc the steam on leaving the boiler 
passes thiough a gioup of tubes foiming a superhcaLci, kc])l hoi by 
the furnace gases, and while taking up heat fioni these tubes ils 
piessuie leniams equal (01 neaily equal) to that in the boiler Super- 
heating is laiely earned fmthei than 100 C and not oflcn so fai. 

72. Constancy of the Total Heat in a Throttling Process. 
An important piopeity of the function /, in any substance, is thai 
it does not change when the substance passes thiough a valve or 
othei constncted opening, such as the porous plug ol' the Joule - 
Thomson expeiimcnt mentioned in Art 19, by which it becomes 
thtottled 01 'wnc-chawn" so that its piessuie chops A piacheal 
instance of this kind of action occius when steam passes lluouoli a 
paitially closed 01 ifice 01 "leducirig valve." Eddies aic louncd in 
the fluid as it uishes thiough the constncted opcnin 5 cind I hcciu-iyy 
expended in foiming them is flittered down intohcataslheysnbsidc 



is U 



To pio\ e that / is constanl in such nn opera I ion we shall consider 
what happens while a unit quantity of the subsUmcc passes Duonuh 
a constncted opening (as in fig 13), and, lo make the mallei clem, 
imagine this unit quantity to be scpaiated horn I he iesl ol (he 
substance by two fnctionless pistons, one of which ( /) slides in NIC 
pipe that leads to the constiiction and Hit ollui (ft) slides in (he 
pipe that leads away fiom it On one suit , as Die substanee comes 
up, let its piessuie be P l5 volume Fj and mtmuil cneigv J^ () 
the othci side, aftci passing the constiiction, lei its piessmc b< /', 
volume F s and internal eneigy E z As cacli po, l lO n appioael.( s I \u 
constiiction, ^oikis done upon it bv Die subslance behind pushing 



I PROPERTIES OF FLUIDS 75 

the imaginary piston A, and the amount of that work done 
hilc unit quantity is passing is P 1 V l After each poition has 
issed the constnction it does work upon the substance in front 
T pushing out the imaginary piston B, and the amount of that 
3rk is P Z V 2 for tlic whole unit quantity. Any excess of the work 
me by the substance on piston B over the woik done upon it by 
ston A must be supplied by a reduction in its stock of mteinal 
ergy. Hence 



sm which #> + AP Z V 2 = Ej + APJ\, 

ins the total heat does not change in consequence of the thrott- 
ig. The imaginary pistons Aveie introduced only to make the 
asoning more intelligible, the argument holds good whethei they 
e there or not It applies to any fluid, and to any action in A\ Inch 
eic is a fnctional fall o( pressure 

We might accoidingh describe thequantiU J as that piopeiiy of a 
b&tanct' which doa> not change in a thiotthng pjoccss l . 

73 Entropy of a Fluid. In reckoning- the entiopy of a fluid 
_' foil on the sume comcntion as in reckoning internal eneigy the 
itropv of lire liquid at C is taken as zero Consider, as before, 
pioccss in which the liquid is first heated undci constant pressure 
id then vapoiized at that pressure During the heating of the 
[ind fiom an initial tenrpeiature T n to am tempeiatuie T (on the 
>solulo scale) the entiopy incieascs b} the amount 

f r dQ T adT 

\T~T ~ T a T ' 

heie a is the specific heat at constant piessure. 

If a could be treated as constant this Avould give on integration 

a (log. T- log. T ) 

In the case of Avatcr cr is not far horn constant and equal to unity, 
cncc a rough value of the entropy of water <, at any temperature 
is given by the expicssion 

loo t T - loo, 278 

* It is assumed that no heat is taken in or given out and also that the velocity 
the pipes is so small that no account need bo taken of any chftcience m the kinetic 
oiffy of tho slion.m m the pipes befoie and after passing the constnction, once 
D eddies have subsided If the stream has acquired an appreciable amount of 
ictjc eneigy aitoi the piocess, there will bo a coirespondmg reduction in / (See 
I 104) 



76 THERMODYNAMICS fen. 

Moie accmate values of cf) w aic obtained by nsin<> a foinmla 
devised by Callcndai which will be given when the deuvalion ol In*, 
tables is described (Chap VIII). In the tables Ihcic is n column 
foi the entiopy of watei at vauous tempcratmes, Ihc piessiiu in 
each case being the satuiation piessiuc at that lempcialme. It is 
the amount of entropy which the uaLei has at the end ol Ihc liisl 
stage in Art 56. when steam is just about Lo be 1'oimod 

During the second stage an additional amount of heal L is I;ik< n 
in at constant temperatuie T s , namely the Leni])cuiliiio .il uhich 
steam is formed under the given picssuie Hence ih< cnhopy 

increases by the amount ^- , and we have, foi Lhc cnhopy ol 

* 4 

satiuated steam, 



Values of <^ s aie gi\ en in the tables 

Dining supei heating theie it> afiuLhei uicicase ol cnliopy .is I lie 
substance takeb in moie lieat. The cntiopy ol snpei healed slc.un 
at vauous piessuies and tcmpeiatuics Mill be found in OIK ol llu 
tables. It can be calculated by means of a foinuila \vlnch \\ill be 
given latei 

74 Mixed Liquid and Vapour. Wet Steam In in my ol flic 
actions that occiu in steain-cnginct, and iclnyi inlinn machines uc 
have to do not with diy saLuiaLcd \apom bill \\ilh a imxhiic 
of satiuated vapoiu and liquid In the e\liiulci ol a sIcani-cMnmc, 
foi example, the steam is generally wet, it contains a piopoi lion ol 
watei which^anes as the stioke piocccds. Whuian\ such niivlim 
is in a state of theimal equihbiium the liquid and \apom li i\ c I IK 
sametempeiatuie, and the vapoiu is salinalcd. \Vlial is c,ill< d lh< 
diyness of wet steam is mcasmcd by Lhc fiacfion q ol vapoiu ulu< li 
is present m unit mass of the mixLiue When Ihc diyncss is known 
it is easy to deteimme othei quanhUcs Thus, Kckomng m c vci> 
case pei unit mass of the mixluic, we have 

Latent Heat of wet steam = qL = q (/., ~ J ttl ) ( [ ), 

Total Heat of wet steam, I q = I w + q L - / s - ( I < { ) L (a), 
Volume of wet steam, V, L = qV, + (1 - f/ ) V m (,{), 

\\hichibveryneailj equal to qV B unless Ihcmivlim is so \ u | 
as to consist mainly of watei , 

Entropy of wet steam, ^ = ^ + ( ^ ^ _ ^ ~ '/) L ( , } 

*- S -L 



j j. J-^V^JL J.JJH.J. j.J-ikJ VJL JL'JLiUlJJ'D 77 

lorn (2) it follows that when the total heat I a of wet steam is 
lown, the dryness may be found by the equation 

T T 

r , _ ts. 1 (K\ 

1- T _r ' ()- 

L a -* , 

>mbinmg (2) and (!), and eliminating <?, we have 

I, = / + T s (</, - w ) (6), 

Inch is a com ement expiession foi iindmg the total heat of wet 
jam when Lhc data aie the tempeiatine and the entiopy. An 
teinativc foim is 

jQ=J.-Z\(k-&) . . (7). 

these t:\picssions /, is the total heat of water, and /, that of 
y sal united steam, al the tempeiatine of the wet mixtiue. 
All these fonnulas apply to a mixtuie of aii3 r liqiud with its 
pom 

75 Specification of the State of any Fluid We have now 
oken ol the follouin^ quanLiLics, \\hich aie functions of the state 
the substance Thcv all depend on the actual state, not on how 
dt sLilc h.is been leached 



The tempt Kitiue, T 
The pitssinc, P 
The volume, V 
The InU'uuil EIKIUN 



These foiu aie icckoned pei unit 



The ToUil IIe.il, / j quautitv ol the j, u butane c 

The Enliopv, f/> J 

A subslanee may change its state m main dillcient \\a\b it may 
inslanee take in heal al constant volume or ^hile expanding, 
ma\ expand 01 be computed Mith 01 without taking m heat, 
pansion ma\ lake place tluouyh a Ihiottle-vah e 01 nuclei a 
,Lon BnL in any change of state whatcvei, the amount by which 
jh of these quantities is alteicd depends only on what the initial 
d final slates aie, and not at all on the paiticulai piocess by 
uch the change of stale has been effected 

Plicic aie olhei quantities, such as the heat taken in, 01 the work 
nc, which depend on how the change of state has taken place, 
dealing with them we have to distinguish between one piocess 
change and anolhei, even when both piocesses bung the sub- 
nice fiom the same initial to the same final condition. 



78 THERMODYNAMICS n-n. 

The woikmg substance may be a liquid, 01 a mixluie of liquid and 
vapour, 01 a diy-satuialed 01 supeiheated vapoiu. The condition 
of a dry-satmated vapour is only a boimdaiy concliLion between 
that of wetness and that of supeiheat. To specify completely llu 
state at any instant it is enough to give eithei the piessuu 01 I lie 
tempeiatuie and one of the othci foui quantities named in Mils lisl. 
Thus if P and V are gnen the state is full}'- defined all I he ollici 
quantities can then be dctcimmed, pioMdcd, of comsi, wi- IUM- 
sufficient experimental knowledge of Lhe chaiaclcuslics ol 1 I he 
substance. Oi we may specify the state by giving auolhei pair ol 
quantities, such as T and <f> 01 P and /, 01 </> and /. 

Moie Generally, any two of these six quantities will sei\ r as d.tf.i 
in specifying the state, so long as the substance is homogeneous, 
but when the substance is a mixluie of liquid and vapoui [hi 
piessme and tempeiatuie do not suffice without some olhci pai- 
ticulai such as the diyness q, 

With legaid to these functions it may be useful Lo icpeaL lic-ic 
that 

T is constant in isotheimal expansion, 

</> is constant in adiabatic expansion: 

/ is constant in expansion thiough a tluotllc-vah c or poious plug. 

76. Isothermal Expansion of a Fluid Isotheimal Lines on 
the Pressure-Volume Diagram. A saturated vapoui can expand 
isotheimally only uhen it is wet the pioccs.s conesponds to I IK 
second stage in the expenmcnt of Ait. 56, it goes on nl conslanl 
piessme and invohes change of pait of the liquid in llu \vi I 
mixtme into vapoui Sinulaily, isotheimal com])ics,su,n ol .1 we I 
vapoui imolves condensation of pail of it Isolhcimal IIIH-S on llu- 
piesbure-volume chagiam foi a mixture of vapoiu and liquid ai 
straight lines of unifoim piessme. 

It is instinctive to considei the genoiul form ol (lie isollimn.il 
lines as the fluid passes successively thiouoh the sl.j-.i-s ol Ixn,.. 
(1) entnely liquid, (2) a mixtuie of vapoui and hquul^;}) ( ul nHv 
vapoious, by having its piessme giadually icducid iimiu- con- 
ditions such that the temperature lemains eonslanl lluounho,,[ | h, 
piocess Imagine for instance a cylmdei to conla.n a q,unl.ly of 
the liquid undei piessme applied by a loaded pjslon, mid le-l (he 
cylinder stand on a body at a definite constant Icnipcial.uo wluc-l, 
wU supply enough heat to it to maintain the lempei.-Uuu- un- 
changed when the pressme of the piston is giaduullv iclaxed and 



n 



PROPERTIES OF FLUIDS 



79 



the volume consequently increases. Starting from a condition of 
veiy high pressuie, say at A (fig. 14), when the contents of the 
cylinder are wholly liquid, let the load on the piston be slowly 
i educed so that the piessure gradually falls. The contents at first 
remain liquid, until the pressuie falls to the satuiation A^alne for the 
given tempeiature, namely the piessuie at which vapour begins to 
form. Thus we have in the pressme-volume diagram a line A 1 B 1 
to icpiescnt what happens while the pressuie is falling during this 
fust stage, the contents are then still liquid. The volume of the 
liquid inci eases, but only ACiy slightly, in consequence of the 
piessuie being iclaxed, and hence A 1 B l in the diagram is ncaily but 
not quite vertical At B t vapoiu begins to foim, and continues 
Conning until all the liquid becomes vapour This is icpresented 
l>3 r B-iC-t, a stage clmmg- \diich there is no change of piessuie At 
(\ the ic is nothing but satuiatcd -vapoui Then, if the fall of 
piessuie continues, a line C 1 D i 
is ti.ieccl, the piogicssne lall 
>!' piessuie being associated 
with a piogiessivc incieasc of 
.olumc The tcmpeiatuie, by 
issiunption, is kept constant 

hiongliout. At I), , 01 at any 

)omt beyond C lt Lhc \apour 

MS become supciliealcd, bc- 

ause Us piessuie is loxvei than 

lie piessuie coiiesponding to 
.ahiralion, and hence Us lem- 

)ciatme it, highci than the 

empcialme coiicsponchng to 

almahon at Ihc actual pics- 

uie Anysuch line^/ZJC'Zhs an 

solhcimal foi the substance in 

he successive states of liquid 

A lo B), liquid and vapom 

nixed (B to C), satuiatcd 

'apour (at C), supei heated vapoui (C to D) Now take a much 

nghci tempeiature We get a similai isotheimal A Z B 2 C 2 D 2 , and 

it a still highei lempeiatme anothei isotheimal A^B 3 C 3 D S , and 

o on. The highei the tempeiature the neaier do B and C 

ipproach each othci, and if the temperature be made high enough 

he horizontal portion of the isothermal line vanishes. 




VOLUME 



Fig 1 1 Isollioj nial Lines 



77 The Critical Point: Critical Temperature and Critical 
Pressure. A curve (shown^by the biokcn line) duiwn thiou<>h 
5^,63, etc. is continuous with one passing tlnough (\(\C^ and 
it is only within the region of which this cm vc is the nppei boundary 
that any change fiom liquid to vapoui taLes phcc. The bunch 
B 1 B 2 B 3 , which shows the volume of the liquid, meets the Inaneh 
C^Cj, uhich shows the volume of the saluraled vapour, in a 
jounded top. The summit of this cui\c icpicscnis a slate wlucli 
is called the CnticalPomt The tcmpeiatmc for an isolhcinial line 
E that would just touch the top of this cmve is called the Cnlicul 
Tempeiatuie We might define the ciitical Icmpeiatmc in miolhei 
way by saying that if the tempeiatuie of a vapour is above llu 
cntical tempeiatuie no piessme, howcvei gical, will cause it In 
liquefy. The piessuie at the critical pond is called the Ciitnul 
Piessuie, at any higher piessuie Lhc substance cannot exist as .1 
non-homogeneous mixture, paitly liquid and paillv v.ipom. 

Staiting fiom D and mci easing the picssme, Lhc tenipeialuu 
being kept constant, ^e mav tiace anv of the isolhemuils luck- 
waids The initial state is then that of a gas (asupcihealed v.ipom) 
If the tempeiatuie is low enough \\c have a discontinuous pnx iss 
DCBA as the piessuie incieases C is i cached whin the \apoui is 
satuiated and condensation begins at B condcnsalion is eoiupkU , 
and fiom B upwards towaids A we aie compicssing hrjiiid A I any 
point between C and B the substance exists in two si, -i Irs ol 
aggiegation, pait is liquid and pait is vapoui But it UK- hm- 
peiatme is above the cntical tempeiatuie the isoMu-inul is OIK 
that lies altogether outside of the boundary curve, shown by llu- 
biokenlme, m that case the substance docs not siift'ei ,my sluup 
change of state as the piessuie uses. It passes fiom the sl.ilc' of 
a gas to that of a liquid in a continuous mannci, tollowmo a ( . ()IIIS( . 
such as is indicated by the lines F or G, and at no sl,,"^ m ,| u . 
piocess is it other than homogeneous 

The critical tempeiatuie foi steam is aboul .'](i;V C , ,md (he 
coiiespondmg piessuie is about 2950 pounds pu squa.e' inch In 
the action of an ordmaiy sbeam-engine the ciideal pou.l is m-v el- 
approached. But with caibonic acid, whose critical lunper.huv 
is only about 31 C , the bchavioui in the nciuhbo.nhood o! the 
cntical point, and above it, is of gieal practical nnpovtanee , 
connection with refngeiatmg machines which employ cai-bome 
acid as working substance. 

Gases such as air, hydiogen, oxygen and so foilh, are vapours 



PROPERTIES OF FLUIDS 



81 



which under ordinary conditions aie very highly supeiheated. 
Their culical temperatures are so low that it is only by extreme 
cooling that they can be bi ought into a condition which makes 
liquefaction possible The cutical temperature of Irydrogen is 
211 C 01 32 absolute. Even helium, the mobt icfractory of 
the oases, has been liquefied, but onl}- by cooling it to a tempeiatuie 
within about 5 degrees of the absolute zeio 

78. Adiabatic Expansion of a Fluid When a satin ated 
/apoui expands achabatically it becomes wet, and if it is initially 
,vet (unless very wet*) it becomes wetter Its temperatme, 
Hcssurc, and total heat fall The fact that its entiopy remains 
maltcied allows the change of condition to be investigated as 
ollows, if we assume that the liquid and vapoui m the mixture 
ue in theimal equilibiuun throughout the piocess. 

For nieatci oenciuhtv we shall suppose the vapoui to be wet to 
>ct>m wilh Let the initial Icmpeiatine be T l and the initial diy- 
iess r/ L In Uus slale Ihe cuhopv is 

t i , <h L i 

9 = ( Pwi + r r 1 

- 1 1 

JY b(ini> Ihe latent heat ol the \apom and <^ x the entiopy of the 
iqmel, bolh al Ihe lempeialme 7\ These quantities aie, tound in 
IK lables Lei llu siibslancc expand adiabaticallv to any loMer 
rnipcMlinc 7\, at \vlueh Ihe latent heat is L, and the cntropv oi 
he. li(|iiul is r/; H)i \u- ha\ c to find the usiilliiiy \ alin ol the di} ness, 
may n<>\\ be e\pn sscd ,is 



nd since Iheie IMS been no ch.'ini;i of entiopy this is equal to the 
nhal value c/; Lfence 

T > 



his e([U.ilion serves lo delei mine the chyncbs attci expansion, and 
nte il is known Lhe volume /'"" is icadily found as in Ait 71 Its 
v.u-t value is q z V^ H- (1 - q z ) F 10j , which is piactically equal in 
rtlmui y cases to g V^ , V^ being the volume of saturated vapour at 

* Whou tho nn^Luio is voty wcL to bogia with, ifc Tbocomes dnei during adiabatic 
panaion, beoauao so muoh of tho portion winch was initially liquid vaponzea under 
o reduced pressure that this moio than makes up for condensation in thepoition 
uoh was initially vapour (HCO Art 100). 

B.T. 6 



82 THERMODYNAMICS [cu 

the temperature T 2 The pressure is the saturation pressure, corre- 
sponding to T 2 Thus the calculation fixes a point m the aeliabatic 
line of the pressure-volume diagram, for expansion from the iiuli.il 
conditions. A senes of points may be found in the same way, conei- 
spondmg to successive assumed temperatures which aie rcaclred in 
the course of the expansion, if it rs desuccl to trace the line. 

In the specral ca^e when the vapoui rs dry and saturatcel to begin 
with, the constant entropy (j> is equal to^., and the expression foi 
the wetness after expansion to any temper atuie T 2 becomes 

As an example of the calculation, let steam initially ehy and 
satuiated at a tempeiature of 190 C (P 1 = 182 1 pounds per sej "V.**- 

inch) expand adiabatrcally to a pressure of one atmosphcic ( Lem- 
per ature 100 C). The entropy, which remains constant duiui<> 
expansion, is 1 5013, </> Wi) is 3119, andI/ 2 rs 539-3. With Lhesc dal.i 
q 2 is 804, 136 per cent of Lhe steam has become liquefied, and Lht 
volume which was originally 2 534 cub ft. per Ib. rs 23 157 cub It. 
after expansion. 

Similarly, if the substance is entncl^ liquid in the initial stale-, 
the picssuic being sufficient to prevent vapour fiom foi mmi>, 
aeliabatic expansion will cause some of iL to vaporize Its initial 
entropy is iw , and since this does not change, 

J- > / i i - 



after expansion Lo a tempeiaturc T z 

Thus, vvjren watei initially at 190 C., and aL the coiicspondm<> 
saLination picssuic of 182 1 pounds pe j i sq. inch, expands adia- 
batically to a pressure of one atmosphere, q, becomes 0151 in 
othei woids 15 per cent, ol the watci vaporizes in consequence ol 
the expansion. The resulting volume is 4-127 cub. ft per Ib 

Conversely, il the wet mixture in this condition wcic compicssc d 
aehabatically it would become wcttci dunng compression, and | 

would be wholly condensed by compicssioir when Lire pressure 
reached 182 1 pounds per square inch. 

An approximation to the foim of the picssnrc-\olume curve 
for the achabatic expansion of wet steam is sometimes obi tuned by 
using an equation of the typcPF 1 " =constanL, and selecting a value 
of the index m appropriate to the initial state. Zcunoi gives foi 7- 

the index in Lhe formula m = 1 035 + Iq wheie q is the chyness 



[J PROPERTIES OF FLUIDS 83 

t the beginning of expansion. But the use of such a method is un- 
atisfactoiy, foi a cm ve whose equation is PV m = constant, starting 
lorn any given initial condition, will agiee with the actual adiabatic 
mve at one other point only. The ciuves cross at that point. 
L value ot m can be selected which will make the equation 
>j7/n _ constant give the light volume when the piessuie has fallen 
om the initial pressure P l to any assigned piessure P 2 For this 
m pose we may wntc 

log PI - logP, 

logF a -lng?Y 

i which F 1 icpicsents the initial volume and F 2 the actual volume 
ached by expansion to P 2 F 2 being dctei mined by the method 
ready given, namely b> fii^t finding the dij-ness and then calcu- 
ting the volume fiom that It will be found that the \ alue of m 
rtained in this way becomes less the fuithei the expansion is 
nied, and also that it is oieatci when the steam is initially diy 
an when jt is Met To take an example, let the initinl state be 
at of diy satmatccl steam at 200 C , loi which the picssine is 
521f pounds pei square inch Adiabatic expansion to \aiious 
wei Lempci.itines and piesbiiies gi\cs the tollowmg lesults 



Tornpoi iliac Piossuio 


Volume 


Index. 




C 


pounds pci MI in 


Divac-js 


(nib feet) 


m 


iLial) 


200 


22524 


1 


2 073S 






190 


18208 


9828 


2 4906 


1 162 




170 


1150G 


9525 


3717 


1 151 



140 52 48 9085 7 400 1 145 

100 14 69 8528 22 849 1 138 

40 1 07 7693 240 35 1 126 

each stage the calculated \ aluc of m is that which would make a 
\o having the equation PV m =P ] F 1 " i pass through the point 
ched at that stage m adiabalic expansion The Zeunci foimula, 
>tcd above, makes the index loi initially diy sic am 1 135, but 
t would make the CUM e he too high in the cail\ stages and too 
r alter the piessuie has fallen below one atmosphcie No constant 
ex can give a leally good appioximation to the actual cm ve. 
!o fill, this Article has dealt with the expansion of satmated or 
, vapom. When a supei heated vapour expands achabatically its 
ansion is divisible into two distinct stages The fust stage bungs 
own to the state of satmation, m the second stage it is a Met 
torn and the foregoing methods of calculation apply. Callendar 

62 



84 THERMODYNAMICS [en. 

has shown, that, m the firsL stage, supei healed steam expanding 
adiabatically follows closely the equation 

p(V - 6) 13 = constant, 

wheie b is the volume of watei at C , namely 010 cubic i'L. 
per Ib Except at high pressuies b is negligibly small compaied 
with V, and may be omitted without serious erior. This equation 
applies down to the point at which the s Learn becomes saturated. 

The amount of adiabatic expansion which will bung a supci- 
heated vapour down Lo that point is detci mined fiom the fact that 
the entropy is constant. We have only to find at what Lcmpciatmc 
(or piessure) the cntiopy of saturated vapoui is equal Lo that of I he 
supei heated vapoui in the given initial state. This companion is 
icadily made when tables 01 charts are available giving the pio- 
perLies o the substance in both states, supeihcaled and satin. iLcd. 
Callendai's tables give the necessaiy data foi steam then use in 
such calculations will be illustiatcd in the nevt chaptci The ch.uls 
which will ako be desciibcd there seive wcllfoi the examination ot 
cases in which the vapoui is supei heated bcfoie expansion. Such 
casei. occui fiequently in steam-engine pi ac I ice With a suitable 
chait it is easy to tiacc the whole coin^e of any adinbaLic expansion 
tlnough the legion oi biipeiheat. past the point of satin alum, and 
finally in the legion of wetness 

It must not be supposed that the expansion of blcnm in an ad ual 
engine is adiabatic, foi there is always some Iransfci of hc'.il be- 
tween the working fluid and the metal ol the cyiindei and pislon 
If it weie practicable to use a peifectly non-conducting male ual 
foi the suifaces m contact with the steam, the ideal of adia- 
batic expansion could be icalizcd. It is appicmmaLcd lo in cases 
wheie the action occins too fast to allow any considerable h.mslVi 
of heat to take place 

Sudden, and theielore practically aduibatic, expansion luun a 
high piessuic may be used to produce a veiy low tcmpcialmr ll 
was in this way Lhat ga,set, such as oxygen and nitiogen wcic loi I lit, 
first time liquefied The gas was compicssccl and was cooled in I he 
compiessed state to a fanl3 r low tcmpciatuie. It was then suddenly 
expanded, and the further cooling which resulted fiom I his ex- 
pansion caused a poition of it to become liquid. 

79. Supersaturation In discussing adiabatic expansion we 
have assumed that theie is at evo^ step in the expansion a con- 
dition of equilibrium in the fluid, that is to say equilibrium between 



[] PROPERTIES OF FLUIDS 85 

tie part that is vapour and the part that is liquid. But it is known, 

s a icsult of experiment, that when a vapour is suddenly cooled 

y adiabatic expansion the condition of equilibrium is not leached 

t once. Suppose the vapour to be initially dry and saturated : on 

xpansion a pait of it must condense if eqmlibimin is to be eslab- 

shed. This condensation takes an appieciable time, it is a suiface 

henomenon, taking" place partly on the inner surfaces of the 

Dntainmg vessel and paitly by the growth of diops thioughout 

ic volume. Consequently the sudden expansion of a vapour may 

roduce tempoiaiil}'- a condition that is called supei satui ation, 

i which the substance continues foi a time to exist as a homo 

:neous vapour, although its pressure and tempeiatuie aie such 

lat the condition of equilibrium would lequne a pail of it to be 

nidenscd In this supersatuiated state the density of the vapour 

abnoimally high, higher than the density of satuiated vapoui at 

ic actual picsbiue The temperature is also abnoimally low, lower 

tan the tempeiatuie of satuiation at the actual pressuie for this 

ason the supeisatuiated ^ apom might be called supercooled The 

ipersatmated condition is not stable it disappeais through the 

mdensation of a pait of the ^ apom, and the lesulting mixtuie of 

ipoui and liquid has its tempciature laiscd In the latent heat 

Inch is given out in this condensation We shall sec latei, in 

nncction with the theory of steam jeti. (Ait. 135), that cx- 

msion involving supcisatmation may occui under piactical 

ndihons*. 

The supeicoohng of a \ apom without condensation is analogous 
the supei cooling of a liquid without crystallization In both 
scs thcic is a clcpailiuc horn I he slate of cqmlibinim, and in both 
scs the icstoialion ol cqiiilibiium invokes an nieva stble action, 
thin I he substance The noimal adiabatic expansion of a \ apoiu, 
alt with in Ait 78, is icvcisiblc, but if theie has been super- 
olmg theie is an iue\ eisiblc development of heat within the fluid 
' An mtoiostmg oxnm[>lo of supeisaturation occms when dust-fieo air saturated 
Ii water vapom is suddenly expanded So long as paiticles of dust are piesent 
nsfc foims (on slight expansion) by the condensation of water on them as nuclei, 
r if they ato lomovod bofoio such an expansion the nifst does not form and the 
>our becomes supoisatmatod If however the ratio of expansion is laigo, so that 
re is much supeicoohng, a mist forms even in the absence of dust in that case 
ppoais that diops of the liquid form about smaller nuclei; which ate not of foicign 
ttor, but prob.ibly consist of gioups of molecules accidentally linked in the course 
ho moloculai collisions that occui m any gas (See Aitkcn, Tians RS E vol 30, 
Nairn a, March 1, 1888 and Fob 21, 1890 also C. T R Wilson, Phil Tiam 
S. vol 189, 1897 ) 



SO THERMODYNAMICS [en. 

when the supercooled vapour passes into the stable state of a 
mixture of liquid and saturated vapoui. 

Duung supeicoolmg by adiabatic expansion steam expands 
according to the formula P (V b} 13 = constant. The foimula is 
the same as for the adiabatic expansion of supei heated steam. IL 
applies whethei the steam be initially superheated or saturated, 
and continues to apply so king as the steam- expands in a homo- 
geneous dry state, as a icsult of supei saturation. 

80. Change of Internal Energy and of Total Heat in 
Adiabatic Expansion "Heat-Drop." When a fluid expands 
acliabahcally from any condition a to any othei condition b Hie 
decieasc of internal eneig}" E a E b is equal to Ihe thcimal cqmva- 
lent of the woik done in the expansion This is because it takes in 
no heat and consequently the 
work which it docs in expand- 
ing is done at the expense of its 
stock of internal cneigy. 

Refeiung to the piessurc- 
volume diagram (fig. 15) the 
woik done dining expansion 
fiom a to b is measuicd by the 
area inabn, consequently in _ 

adiabatic expansion Volume 

T-I n , , 7V T^Jff JO 

E a Ej, = A (aiea wnbu} 

Further, the decieasc of Lolal heat which the substance umlu- 
gocs din ing the pioccss is equal to the Ihcnnal cquiMilcnl of I lie 
aiea cabf To pio\e llus, \vc have, by the definition of the tol.il 
hqat(Ait. 07), 

r _ J7 I ///> y 

1 ti ~ lj a r -" ' j 

and 7 ft =/(l 6 + JZ' t r 6 , 

fiom Avhich 




m 



= A (aiea mnbn -i- aiea camo a 
= A (aiea cabf) 

This is tnic whatever be the condition of the fluid befoi c expansion 
it applies for example to superheated as well as lo satin a bed or wet 
steam, 01 to any gas 



PROPERTIES OF FLUIDS 



87 



It may be instructive to the sbudent to have the same pi oof put 
in a somewhat different foim. From bhe equation winch defines the 
total heat I in any state, namely, 

I = E + APV, 

\ve have by differentiation 

dl = dE + Ad (PV) 

= dE + APdV + AVdP. 

Bub in any small change of state it follows from the conseivafcion 
of energy that the mcicase of internal energy plus the woik done 
by the fluid is equal to the heat taken in, or 

dE + APdV = dQ, 

vvhcie dQ is the heat taken m during the change. Hence in any 
small change of state rfj = ^ + AVdP. 

In an adiabatic opeiation c/<2 = 0, and hence in that case 

HI = AVdP. 

Therefore if the fluid expands adiabatically from state a to state b 
the resulting dccicase in its total heal, namely 



/-/ = A 



VdP. 



This integial is the aica cabf of the picssuie-volumc diagiam 
'fig 15) It is the \\hole \\o\k done m a cslmdcr when the lluid is 
admitted at the picssuic corresponding lo state a, then expanded 
ichabatically to slate b, and Hun disehaiged at the pitssiuc conc- 
ipouding lo slale b 

The decrease of total heat m expansion, I a I b , is called 
Ihe "Ilcat-dio]) " It is a quantity of much importance in the 
lluoiy of heal -engines The aboye equation sho\\s that under 
idiabatic conditions the whole work done in the cyhndci, when 
expressed in heat units, is measured by the heat-drop In the 
next chapter this principle will be applied to infer from the heat- 
drop the work that can be done in steam-engines under \anous 
assumed conditions, and it will be shown how to calculate the heat- 
drop which occurs in adiabatic expansion from anj initial state 



CHAPTER III 
THEORY OF THE STEAM-ENGINK 

8 1 Carnot's Cycle with Steam or other Vapour foi Work- 
ing Substance. We aie now in a position lo study (lie .id <>l 

a heat-engme employing watci and steam, 01 any olliei li([iii(l .md 
its vapoui, as the ^oiking substance To siiiiplil'v (lie lusl coii- 
bidciation of tlie subject as fai as possibk, hi il be suppose d (ha I 
we ha\ e, as befou, a long o^Inuki, composed ol nou-conduel m^ 
matenal except at the base, and filled \\ilh a ii<>n->iidiiel inn 
piston. albO a souice of licat A at some lunpt i.il me 7 1 , , .1 K eei\ 1 1 
of heat, or as we may now call it, a comic usci, (', al some lo\u i 
tempeiatuie T 2 , and also a non-conduclmi> cm ei II (.is in Ail .><>) 
Then Cainot's cjcle oi opeiations can bo pulonmd .is l<>||o\\s 
To fi\ the ideas, suppose that then, is uml m.iss <>l w.ilu in llu 
cyhndci to begin with, at the tempeuilim 7', 

(1) Apply A, and allo^ the piston lo use ai>amsl (In c-onsLml 
pleasure P l which is the satmalion piessmc coiiesponduin | ( , UK 
tempeiatuie T T . The water will take m he. it .md IK eonxdUd ml<. 
steam, expanding isotheimalh .il tlu lempc lalme 7', Tliisjt.iil ol 
the opeiation is sho\Mi by the line ab in hi- 10 

(2) Remote A and appl\ B Allow the cvp.msiou |> , |,,,i, 

adiabatically (to), with falling pics^me, unlil Ihe U mpi i. ,(,i l( |.,|| s 
to T 2 The piessmc will then be P 2 , nmncly, llu piessnn \\hi< h 
coiiesponds in the steam [able to 7\, uhich is (lit l(miKi..lun ,,l 
the cold bod\ C 

(3) Re-motel?, apply C, and compiess Sle.im is eondens. d l,\ 
lejectmg heat to C The action is isolhcuiu.1, and ||, ,,,,SSI.M 
lemams P a Let tins be contmued unhl ., c-c,l,m, p.,,,,1 ,/ ls 
leached, hich is to be chosen so th.,1 ad,ab,,lie eonmu ss,<,n ,|| 
complete the cycle 

(4) Remove C and apply B Continue the compiess,,,,, win. I, 
is no adiabatic If the point d has been noluly c-hoscn, || lls n,|| 
complete the cjcle by ^stoung the woikmg || llld , () lhc sla|r <)( 
Mater at temperatuie T T . 



r. in] 



THEORY OF THE STEAM-ENGINE 



89 



The mdicatoi diagiam for the cycle is drawn in jfig. ifi, the lines 
and da having been calculated by the method of Ai t. 78, for a 
irticular example in which the initial piessnre is 90 pounds per 
nave inch (2\ = 433), and the expansion is continued down to the 
essuic of the almospheie, 14 7 pounds pei squaie inch (T 2 = 373). 




Fig Id Gi mol's r\clo \\illi Nfitei and steam lui \voikiny mibslanco 

iincc I lie pmcess is K \usible, and since heat is taken in only at 
and ic|cclcd only at 7\, Ihc efficiency (by Ait. 38) is 



c lie.it taken in pei unit mass of the liquid is L 1} and theici'oic 



woik clone is 



T 



^sult which mny be used to check the calculation of the lines 
Lhe diagram by compaimg it with the aica, which they enclose 
.vill be seen that the whole opciation is stnctly icveisible in the 
rmodynamic sense. 



90 THERMODYNAMICS [cu. 

Instead of supposing the working substance to consist wholly of 
water at a and Avholby of steam at b, the operation fib might be 
taken to repiesent the paitial cvapoiation of what was originally 
a mixture of steam and watei The heat taken in would then be 
(?& ~ <Ja) ^1 an d as the cycle would still be reveisible the aica of 
the diagiam would be 



82 Efficiency of a Perfect Steam-Engine. Limits of Tem- 
perature If the action heie descubcd could be reali/cd ID 
piactice, we should have a thcrmodynamically pcifcct slcjun- 
engine using sarmatcd bteam. Like any other pcifect heat-engine, 
an ideal engine of this kind has an cmcicnc} r which depends upon 
the temperatures between which it works, and upon nothing else 
The fraction of the heat supplied to it which such an engine uould 
convert into woik would depend simply on the two Icmpcialmrs, 
and Iherefoie on the picssurcs, at which the steam A\as pioduoecl 
and condensed icspectively. 

It is inteie.stmg theicfoie to considci what me lhc hmils of 
tempeiatuie between which s team-engines may be made to \\oik 
The tempeiatuie of condensation is limited by Ihe considcinhon 
that theie must be an abundant supply of some substance lo 
absoib the lejected heat, watei is actually used foi this puiposc, 
so that T 2 has fonts lowei limit the tempeiatuie of the available 
watei -supply 

To lhc highci temperature 2\ and picssuie />, a piadical linn I 
is set by the mechanical difficulties, with icgaid lo sticnglh and 
to lubrication, which attend the use of high-picssmc steam In 
steam motor-cais piChSiues of 1000 pounds pci sq inch have been 
used, but with engines and boileis of the ouhnaiy consliuchon 
the prcssmc langcs fiom about 300 pounds pei sq inch clowns ,uds 

This means that the upper limit of tcmpcrahue, so fai as satin - 
atcd steam is conccincd, is about 215 C A slcam-cnginc, Iheie- 
forc, undei the most favouiablc conditions, comes very I'm shoit of 
taking full advantage of the high tcmpciatinc at which lu<U is 
pioduced in Ihe combustion of coal Fiom lhc thcimodynanuc 
point of view the woist thing about a slcam-cngme is the me- 
versiblc diop of tempeiatuie between the combustion-chambei of 
the furnace and the boiler The combustion of the fuel supplies 
heat at a high tcmperatuie but a gieat part of the convertibility 



in I THEORY OF THE STEAM-ENGINE 91 

of that heat into woik is a I once sacrificed by Ihc fall in temperature 
which is allowed to take place before the conveision into work 
begins. 

If the tempcratme of condensation be taken as 20 C., as a lower 
hmil, the cflieicncy of a pcifcct s team-engine, using satmatcd 
slcain and following the CainoL cycle, would depend on the value 
of I\, the absolute picssmc of production of the steam, as follows: 

Pcifect steam-engine, with condensation a,L 20 C., 
P t in pounds per sq inch being 50 100 150 200 250 300 
Highest ideal eflicicncy - 2SS -330 355 373 384 399 

These numbers cxpicss what fiaclion of the heat taken in by 
the woikmg substance would be comcitible into woik under the 
ideally favouiablc conditions of the Cainot c^ clc 

But it must not be supposed that these \dlues of the eflicicnc}'' 
.no actually attained, 01 aic even attainable Many causes con- 
spue lo pie\cnt sic.un-engines fioni being tlicimodynainically 
|)ci led, and some of the causes of nnpti (ctlion cannol be iemo\ cd 
Tlu-sc nmnbeis will seivt, howe\ci, as one srand.ml of companion 
in judging of I lie pc i Joimancc of ae( n.il engines, and as illiistiating 
the .uhanlage of Iiigli-j)iessuie slcam fioni Hie lliennod} nanuc 
])omlof\nw. We shall see m Ai I 87 that llicie is anolhu .slanchud 
wilh winch Ihc pcil'oinianee ol a ical sicam-cnginc may moie 
appropiulely be complied 

83 Entropy-Temperature Diagiam for a Perfect Steam- 
Engine The nnagmaiv sk.mi-engmc of Ait. SI lias the same 
vei y simpk enliopy-lenipcialiiic di.igiam as anv olhci engine 
which follows C'ainol's C) cle Tlie fom opcialions aic icpiescnted 
by the fom sides of a leclanglc (fig 17) The fust opeiation 
ch,mg( s watci (at the upper limit of tcmpcratmc) into saturated 
slcnm at Ihc same tcmpciahu-e, the enhopy accoidingly changes 
Iiom (/}, to <:/>,, This is shown by the conslmit-lempcia,tuic line 
fib in Jig 17 In the second opeiationAvhich is adiabatic cx- 
pansion the cnlio})y docs not change, and the tempcialurc falls 
lo the lowci limit, at which heat is to be icjccted this is repic- 
sented by the line of constant cntiopy be In the thud operation, 
cd, the Icmpeiatme jcmams constant and the cntiopy is icstoied 
lo its oiigmal value, heat being rejected to the cold body. In the 
fourth operation which is adiabatic compiession the entropy 
does not change, and the temperature rises to the upper limit. 



92 THERMODYNAMICS [en. 

the substance has returned to its initial state in all lespects. In 
order to be compaiable with 
other diagiams which will folIoAV, 
fig 17 is sketched for a pal ticulai 
example in which P : is ISO pounds 
per sq. inch, and P 2 is 1 pound 
per sq. inch: consequent!}' Z : is 
189 5 C and t z is 38 7 C g 

Expressed in tei ms of entropy, "J5 
the heat taken m (during ab) is ^ 
TI (4> s <f>w)- Thisisicpiescnted 
by the aiea undei ab measured 
down to the absolute zeio of 
temperatme, namely the area 
inabn. The heat i ejected (dm ing 
cd) is T 2 (^ (j> w ) and is icpie- 

scntcd by the aiea ncdm. The ff ' Entropy n 

theimal equivalent of the woik Fig. 17 

done in the cycle is accoidmgly 

(T l To) (</> s cf> lu ), and is icpiescnted by the area abed, enclosed 
by the lines which icpiesont the fom icvcisible opciations. The 
efficiency is ^ _ T ,) (fa - <ftj _ I\ - T. 2 

In the example foi which the diagiam is diawn, with Ihe dala 
stated above, the numciical value of this is 326. 

84. Use of " Boundary Curves" m the Entropy-Tempera- 
ture Diagram. In fig IS the diugicim of /ig 17 is diawn ovc-i 
again, with the addition of a cuivc Ihiough a which icprcsenls Ihe 
values nl vanoiis tcmperatuies of </>,, the cntiopy of watei when 
steam is just about to fonn, and a cuivc Ihiough b which icpie- 
sents at various lempciatuics the value of </>,, the entiopy of cliy 
satiuatcd steam These curves aie called Bouuddiy Cmvcs They 
aie icadily drawn fiom the dala in the steam tables Any point on 
Ihc boundary curve thiough a would relate to the cntiopy of water, 
between the two curves any point in Ihc diagiam relates to a 
mixtiuc of watei and steam, to the right of the boundary curve 
Ihiough. b any point would i elate to steam in Ihc supcihcatcd state. 
We aie not at picscnt concuncd willi the outlying regions but only 
with the space between the two curves, willnn which the points c 
and d fall Let the line cd be pioduccd both waj T s to meet the 



Ill] 



THEORY OF THE STEAM-ENGINE 



93 





I- 



boundary curves in e and s. Then the ratio of cs to es represents 
the fraction of the steam which becomes condensed during the 
achabalic expansion be from the condition of saturation at b 

To prove this we may first consider the meaning of any hori- 
zontal (isollicimal) line such as se on the cntiopy-tempeiatnie 
diagram between the two 
bomiddiy curves It ic- 
picsents complete con- 
densation of 1 Ib of dry 
satuiatcd steam, under 
constant tempeialiuc 
andpicssurc Dining Us 
convcision fiom the con- 
dition of diy saturated 
steam (al &) to water (at 
e] the steam gives out a 
quantity of heat which is 
measured by the aica 
undci I he line, namely 
the aica of>cl Any inter- 
mediate point in Ihc line 
icpiLscnls a mi\line of 
water and steam, llms c 
icpiescnts a mixtmc which, though it has actually been pioduecd by 
adiabatic expansion fiom b, might have been pioduced by puitial 
condensation fiom 6 undei constant pressine, apioecss which would 
be u'prchcnled by AC, 01 by partial cvapoiatiuu under the same 
constant picssuie fiom e, a pioeess which would be icpicsentcd 
by cc. Now if the nnxluie at c, weie completely condensed undci 
couslant pressuie to c, the heal given out would be mcasmcd by 
Ihc aica tied This heat is gi\cn out by the condensation of that 
part of the mixture which consisted of steam Hence the fraction 
which existed at c as steam, or in other woids the diyness of the 
nuxtuic at c, is measured by the latio of the aicas ncel to oscl, 
which is equal to the ratio of the lengths cc to cs Hence also the 
ratio cs to as mcasiucs the wclness of the mixture at c. 

An en tropy- tempera tuie diagram on winch the boundaiy curves 
aic diawn theieforc gives a convenient means of determining the 
wetness of steam at any stage m the process of adiabatic expansion. 
It is only necessary to chaw a vertical line through the point repre- 
senting the initial condition. That line represents the adiabatic 



Entropy 
Flit IS 



94 THERMODYNAMICS [CH. 

process, and the segments into which it dnides a horizontal line 
drawn from one boundary curve to the other at any level of 
temperature represent the proportions of water and steam in the 
resulting mixture. Tins is true not only of the final stage, when 
adiabatic expansion is complete, but of any intermediate stage, 
for the argument gi\en aboA e obviously applies to a horizontal 
line diawn at any temperature between the two boundary curves. 

Srmilarly the point d which represents the wet mrxtinc at the 
beginning of adiabatic compressron da, shows by the ratro of 
segments ds to de what is the proportion of Avater to steam at Avhich 
the third stage of the cycle has to be arrested, in older that adrabatrc 
compressron may brrng the mixture Avholl}' to the state of Avater 
Avhcn the cycle rs completed by the operation da 

The student should compare this graphic method of studyrng 
the A\ r etncss resulting from adiabatic expansion Avith the calcula- 
tions given in Art 78 He Avrll observe that both have the same 
basis. At any temperature T the length es of the isothermal line 
draAvn from the water boundary curve at e to the steam boundary 
curve at s rs L/T, and the intercept cc up to any intermediate point 
c on that line is qL/T, Avhere q is tire di} ness of the mixtmc at the 
point c. The same principle ol course holds for the entropy-tem- 
perature diagiam of any other fluid. 

85 Modified Cycle omitting Adiabatic Compression. Con- 
sider next a modification of the Carnol c\clc of Ail 81 Let 
the fust and second opeiations occur as thc} r do there, but let the 
third operation be continued until the steam is wholly condensed. 
The substance then consists 
of Avatcr at T 2 , and the cycle . _" * 
is completed by healing it, in 
the condition of \\alci, fiom 
jf' 2 to r l\ In the simple 
engine of Ait 81, \\hcre all 
the operations occur in a single k _ e 
\ cssel, this could be done by 

increasrng I he pressure exert- j, J0 . 1{) 

cd by the piston from P 2 to 

P 15 aftci condensation is complete, then icmoA'ing the cold bod} r 
C and applying the hot body A. The Avater is therefore heated at 
P! and no steam is formed till the tcmpcratmc reaches T 1 

The pressure-volume diagram (or indicator diagiam) of a cycle 



Ill] 



THEORY OF THE STEAM-ENGINE 



95 




modified in this mannei is shown by abce m fig. 19 The sketch is 
not chawn to scale As before, ab is the operation of foiming 
steam, from water, at r l\ and P 15 be is achabatic expansion from T 1 
and 1\ to T z and P 2 Then ce is complete condensation at T 2 and 
Po The fouith operation ea now involves two stages, fiist laismg 
the pressme of the condensed water from P 2 to I\ and then heating 
it fiom To to T t . Dining both of these stages the changes of 
volume aie negligible in comparison with those that take place 
in the othci opciations. 

The cntropy-tempciature diagi am ioi this modified cycle is shown 
by abcc m iig 20, whuc the same Icttcis as in fig 19 are used foi 
coiicsponding opciations 
As m the Cainot cycle, 
ab icpicscnls the conver- 
sion of a pound of watci 
at r l\ mlo diy saluialcd 
steam at 2\, and be ic- 
piesenls Us adiabalic ex- 
pansion lo 7\, lesnlling m 
a Met mivlnic at c, Ihc 
diyness of \\hieh is mpa- 
suied by Ihe latio tic/cs 
Then cc upiescnls the 
complele condcnsalion at 
T 2 ol Ihe sleam in llus wet 
inixlnu, and ca, Avhich 
])iaclic.illy coincides with 
the boundaiy ( in \ c, icpie- 
suils llu le-luahng ol Ihe c 

condensed \val< i liom T> to 7'j , al'lci Us picssmc has been uuscd 
lo 1\ so Ihal no sleam is lormccl during Ihis opeialion ' 

The woiking subslnncc behaves rcvcisibly Ihroughout all these 
opciations, and Iheiefoie Ihe woik done in the cycle is repicscntcd 
by Ihe area nine in Ihe cntiopy-tcmpciatuic diagram of fig 20. 
The diagram further exhibits the heat lakcn m and the heat re- 
jected The whole heat lakcn m is mcasmed by the area leabn, and 
of this Ihe area ham mcasmes the heat taken m during the last 

* Tho lino ca in Loth dmgiams, figs 19 and 20, ically stands foi a broken ]mo 
la'a, whoio ca' loptowonts tho rawing of pressure fi om P^ to P : at constant tompera- 
buic TI, and a'a iopicnonts tho heating from T z to T 1 at constant pressure P lt In 
fig 10 a' pmctically comoidos with a, m fig 20 a' practically coincides with c 



m Entropy n 



96 THERMODYNAMICS [CH. 

operation, while the water is being re-heated, and the area mabn 
measures the heat taken dmmg the fhst opeiation, while the 
waber is tuining into steam. The aiea ncel measures the heat re- 
jected, namely dining the condensing piocess ce. 

To express algebiaically the woik done in the cycle, icfer to the 
indicator dmgiam, fig 19, and let the lines ba and cc be produced 
to meet the line of no volume m j and k. Then, by Art 80, the 
aveajbcL is an amount of Avoik equivalent to the difi'cicncc of total 
heats 7. _ / 

-*& * CJ 

namely the "heat-drop" of a pound of steam in expanding adia- 
batically fiom the condition at b to the condition at c The small 
area jaeL is (P t P>) F Wa where V Wn is the volume of a pound of 
watei at T z , which we ma}'- take to be piactically constant ibi the 
piuposes of this calculation 
Hence the expression 

7 6 - I e - A (P, - P 2 ) F B> 

is the thermal equivalent of the woik done in the cycle If figs 19 
and 20 weie both caiefully diawn to scnle foi any paiticular 
example, a measiucmcnt of the enclosed aica abcc in cilhci figuic 
would give a lesult m agioemcnt wiLh this calciilalion. 

86. Engine with Separate Organs The impoi lance of llie 
modified cycle descnbcd in Ait 85 lies in Ihc fact of Us being UK- 




Pjg 21 

nearest appioach to the Carnot cycle that can be aimed at when the 
operations of boiling, expanding and condensing are conducted m 
separate vessels. The imaginary engine of fig. 16 had one organ 



mj THEORY OF THE STEAM-ENGINE 97 

only a cylinder which also served as boiler and as condenser. 
We come neaier to the conditions that hold in piaclice if we think 
of an engine with sepaiate organs, shown diagiammatically in 
fig 21, namely a boiler A kept at T 1} a non-conducting cylinder and 
piston B, and a sniface condenser C kept at T 2 To these must 
be added a feed- pump D which ictiuns the condensed watei to 
I he boilei Piovision is made by which the cylinder can be put 
inlo connection wilh Lhc boilei 01 condenser at will 

With llns engine the cycle of lig in can be pei formed An in- 
dicator eliagiam foi the cylinder 7? is sketched m fig 22 Steam is 
admitted fiom Ihc boiki, giving Ihc line ]b At b "cut-off" occurs, 
lh.il is lo say the vahe winch admits slcam fiom the boiler to the 
cylnulei is closed The slcam in the cyhnclei is then expanded 
aduibalicdlly to Ihc piessiirc of the condenser, gu mg the line be 
Al ( the "exhaust" \alvc is opened vihich connects the cylinder 
with (he condcnsci The pislon then letmns, discharging the 
slc.un lo Ihc condense! and giving the line c/i The area jbcl 



l<V r 22 Fig 22 a 

epuscnls (lie 1 \\oik (loin- in llic eylindu li Tin. condensed wafci 

s Hun icliiuud In Hit 1 boilci bv llu fccd-pmu]), .nul Ihc mehcaloi 

lugiam showing Ihc \\oiL ixpcndtd upon Ihc pump dining Llns 

>pu,ilion is sUilcliul in dg 22 a II is Ihc icclangle hi'tt), when 

c icpicstnls Ihc np-shoki m which Ihc pump (ills with walei 

I the pitssmc /\, and <i) icpicscnls Hie dowu-sliokc in which it 

ischaigts walci lo Ihe boiler agamsl the picssurc Pj If we 

upupose Ihr (lugiuin oi Ihc pump on tluil of the cyhnelci we get 

IK u (lif('cuMic( , iiani( ly alitr (fig 10), lo icpiescnl the net .unoiint 

{ work done by Lhc Hind m the c> clc. It is the excess of the work 

one by the /Imd m Ihe cylinder over that spent upon it m the 

nmp. 

Taking- the two parts sepa lately, the adiabatic heat-drop, 

* ~ -LI 



98 THERMODYNAMICS !<'" 

is the thermal equivalent of Lhc woik done by UK- ilmcl in UK- 
cylmdei, and ^ _ p,) J/ UB 

is the woik spent upon the fluid in the feed-pump Acroidmidy 
the diffeience, namely 

I,-I c -A(l\-PJV Wi , 

is, as befoie, the theimal equivalent oi 1 (he woik obl.umd in Mie 
cycle as a whole. 

87. The Rankine Cycle This cycle is commonly called I lie 
Rankine Cycle Like the Cainot cycle il i-cpu-suils .in uh.il Ih.il 
is not practically attainable, foi it postulates a complete absence 
of any loss thiough tiansfci of heat between llic sU.im .nul I he 
smfaces of the cylinder and piston. But it allouls .1 vriy v.iln.ible 
cntenon of peiformance by famishing a stand, nd \villi \\lueh UK 
efficiency of any leal engine may be compaiod, a slandaid \\lnch 
is less exacting than the cycle of Cainol, but 1. 11101 I'm companson, 
inasmuch as the fouith stage of Ihc Cunol cvcli 1 is luetss.mlv 
omitted when the steam is icmovcd fiom I he cylnuhi Ixlon con- 
densation. A sepaiate condensci is indispensable in an\ u al 
engine that pietends to efficiency 

The use of a sepaiate condensci was in lad one <>l UK HKM! 
mipio\ements \vhich distinguished the sle.un-en^iiK ol W.ill liom 
the eaiher engine of Newcomcn, A\heic I ho sli'.im \\.is (ondt nsi d 
in the working cylmdei itself The mliodnelion ol'.-i s<'p;n,il con- 
denser enabled the cylmdei to be kc-pl comp.iiali\ t-l\ lu>l, and 
theieby i educed immensely the loss lhal h.id o((iiii(d in <,nliu 
engines thiough the action of chilled cylmdei sml.icis upon (In 
enteimg steam. But a sepaiate condonsei, gK.illy Ilion^li il .idds 
to efficiency in piactice, excludes Ihc compu-ssion sl.in* <>l lh< 
Cainot cycle, and consequenlly makes Ilic Hankiiu c\ cli Ihc 
piopei theoietical ideal with which Uicjicifoinicincc ol'.i n-.il ( rmuu 
should be compaied. 

The efficiency of the Rankine cycle is less llun lh.il ol 
a Carnot cycle with the same Imnls ol Icmpi-ialnic Tins is 
because, in the Rankine cycle, the heal is nol all l,ik<'n in a( Hit- 
top of the lange In the Rankine cycle, as m Cm mil's, all Ihc 
intemal actions of the woikmg snbslame ,ue, by assiimplion, 
reveisible, and consequently each elejnenl of Ihc whole hcal- 
supply produces the gieatest possible mech.imcal elTrc'l when 
legaid is had to the temperatme at which lhal elemenl is Inkcn in. 



- 

ml THEORY OF THE STEAM-ENGINE \ ..\ 

J \ -r, \, 

V Ts' 

But part of the heat is taken in at temperatures lowcKtan %\> 
namely while the woikmg substance is having its tcmpGrattfre; f 
raised from T z to 2\ in the fouith opciafcion. Hence the average'- 
efficiency is lower than if all had been taken in at T 1} as it would 
be in the cycle of Carnot. "* 

pound of steam does a laiger amount of woik in the 
cycle than it docs in the CainoL cycle. Tins will be 

t when the uieas aic compaied which lepiescnL the work 
in Llic corresponding diagiams- the aica nbcc with the aica abed in 
lio 20. 13 ut the quantity of heat that has to be supplied for each 
poiuid in the Rankine cycle is also gicatci, and in a gieatei latio 
iL is mcasmcd by the aica leabii, as against mabn Ilcncc the 
efficiency is less in the Rankine cycle One may put Ihe same thing 
in a different way by saying thai, in the Rankine cycle, of the 
whole heat-supply the paitZra/M does only the compaiatively small 
amount of woiktw/, and the lemamdci oC the heat-supply, namely 
ma\)n , does the same amount of woikas it would do m aCainoLc>cle 

88. Efficiency of a Rankine Cycle. Tjkmg in the fiist instance 
a R.nikine cycle in winch the steam supplied to the cUmdci is diy 
and saturated, tlic \\liolc amount ol heal taken in is the quantity 
loquivcd to convcit water ,il Pj and T 2 into satuiatcd steam at 
J\ Tliis quantity is / - {/ 1t , a + A (l\ - P z ) /' u , a }, Jbi the total 
heat oi Iho watci at P l and To is gieatei than l Ui by the quantity 

-/(/*! -^)^ a 

Tlic* woi k done is (by Ait. So) CCMI.I! lo tlic lu.ii-diop muni* the 
woik spc-nf- in the k-td-pmnp, 01 / - / c - A (I\ ~ 1\) l\ tt whcic 
/ is I he Lolal licat of L ho we I mixluie aflu adiabalif expansion 

The* c'/Iic'iLiic'V in I lie c\clc as a \\hole is Lheicfoic 



The ieecl-pump U-im A (P l - P z ) V w , is iclalivcly so small that 

it is of Lei i omiUed in calcuJadons iclatmg to ideal efficiency., jiiht 

is it is omilU-d in staling the icsults of tests of the pcifoimance of 

L-eal ongiiies. In such tests it is customaiv lo speak of the work 

lone per Ib. of steam, without making any deduction for Ihe woik 

hat Jms to be spent per Ib in returning the feed-water to the 

loilcr. 13 ut in the complete analysis of a Rankine cycle the feed- 

>uinp term has to be taken into account, and il is only then that the 

i,rca of Hie entropy-temperature diagram gives a true measure of 

he work done. It should be clearly undei stood that the heab-drop, 



100 THERMODYNAMICS [i it. 

by itself, is not an accurate measuie of the woik done in I he 
Rankine c} 7 cleas a whole, 1101 is the hcat-diop equal lo the ciu-Iosi d 
areaof the entropy- Lcniperatiuc dia^iam, until the thermal equn a 
lent of the work spent in tlioieo'd-pump has been deducted fioin il 
If however we aic concerned only wilh I he \\ork done in the 
cylinder of the ideal engine, then the heal -drop alone has ID U- 
leckoned. It is the exact mcasiue of lhal woik The ralio ol (lit 
heat-drop to the heat supplied shows whal piopoihon of I he supply 
is conveited into woik in I lie cylinder, under the ide.il conditions 
of adiabatic action it is a lalio nearly identical \Mlli (In ellieu nrv 
of the Rankmc cycle, and cvcumoio useful asaslandaid \\ilh \\lueh 
to compare the pciloimancc of a leal engine In (he .iclual pt i- 
formance of any mil engine Iho amounl ol uoik done in the 
cylmdci necc^anly falls shoj I ol Ihe adiabahc heal-diop IK can-. 
the woikmg substance loses some heal lo (he cylinder \\alls. TJu 
extent Lo which it falls shoiL is .1 mallei I'oi (ml, and IUM-( Ih.tl 
has been asceitamcd by luals of engines ol <.|\rn lyp< s, eshm.ih s 
may be made of the pcifoinuncc of an ennine IMH|< i d< SKMI nsni 
the adiabatic hcat-di op as Llie basis of (he c.ileiilalion, \\\\\\ \ -.mi 
able allowance foi probable wasle 

89. Calculation of the Heat-Drop Ft is fheiefou ess.niml 
to be able to calculate the heal-diop in id<al CMIIUS und. i ., M \ 
assigned initial and hnal con- 

ditions. Foi thispiuposenc / \ 

have to find / c , the tolal c/_ ___ , \ . 

heat of wet steam aftci adia- 
batic expansion One A\,iy 
of doing so would be fnsl lo 
calculate the diyncss q and 
then apply equation (2) of 
Art 71, / = /,-]- qL Bui 
equations (6) and (7) o| 
that Article give a more con- 
venient method, which is 
available here because we 
know the entiopy of the " Jg "' 

mixtuie Theseexprcssioiisuia y be(hreell y ol,lauH-(lb V e< ) ns l d ( ., 1 , ) " 

' 



\ 



what amount of heat the wet uuvhn-e would luue lo ,-, u i,l, l{ 
weietobewhollycondcnscd,andwhal amouutof heal ,1 ucu.ld | mxi 
to take up if it weic to be wholly evaporated, urnle, Uu 



in] THEORY OF THE STEAM-ENGINE 101 

pi ess 1 1 re coiiespondmg to the tempeiatuie of saturation Tm eithei 
case. To bung a mixture at c (fig. 23) into the condition of water 
at e would require the lemoval of a quantity of heat equal to the 
area under ec, namely T (<f> - </>J, wheie </> is the entiopy at c and 
tj> m is the entropy of watei (at e}. On the other hand, to bring it to 
the condition of satmated steam would requue the addition of a 
quantity of heat equal to the aiea under cs, namely T (cf) s <). 
Hence the total heal of the mixtuie at c is 

/ = I w + T (</> - &,), 
or 1 C = I S - T (<f> s - <). 

Oi these two cvpic&sions the second is the moie convenient 
because steam tables gcneially give moie complete sets of values 
of (/>, than of </> w 

The entropy cf> of the wet mixture is the constant entiopy under 
which adiabatic expansion has taken place it is to be calculated 
fiom the initial conditions This method of finding the total heat 
ciltci admbatic expansion makes no assumption as to \\hat the 
btale of live steam was befoie expansion it is equally valid \\hether 
the steam was diy, wet, or supeiheatcd to begin with What is 
assumed is that nftci expansion the steam is wet, and that \\i\\ m 
ooncial be tuie even if tlicic be a large amount of initial supeiheat. 
IL is also assumed (Ait 78) that the vapoui and liquid in the 
wet mixtuie aic in cquilibmim 

In the Knnkme cycle ot Arl 87 it \\as assumed that the steam 
-uas diy mid saliualcd al the bc'omning oi I he adiabatic expansion 
Consequently its initial tola! heat was / v and tlnouohout ex- 
pansion was equal lo </> i Under these conditions the total heat 
tiflci aduibaLic expansion is 

/c--^-^(^ B -<W, 

jnd the hcut-diop is 

J r i 1 -J 6 = A 1 -A 1 +T.tf. 1 -W 

To take a numerical example, let the steam be supplied in a dry 
stiLui.'iLcd stale at a piessiuePi of ISO pounds pel squaic inch, and 
lei it expand achabatically to a pressuie P 2 of 1 pound pei squaie 
inch, at which it is condensed. With these data we find from the 
tables '1\ - 402-58, T 2 = 311 -84, ^ = 1 5620, <f> Sa = 1 9724, 
y Si - OGS-53, J, t = 012 10 

Hence the total heat aftei adiabatic expansion to the assumed 
pressure of condensation is 

I = 612 5 - 311 8 (1 9724 - 1-5620) = 484-5. 



102 THERMODYNAMICS [en. 

And the heal-diop, 

/ Sj - / t = 60S 5 - J81 5 = 184 

If \\e considci I lie Rankine cycle as a. whole, the feed-pump 
tcim A (Pi - P z ) V m is 

(iso-|)i44 xooiGi Q 

1400 

Deducting this fioin the hcat-diop we have 183-7 pound-calories 
as the theimal equivalent of the net amount of woik done in Lhe 
Rankine cycle. 

The heat supplied is 

7 6j _ 2 wt - A (P l - P 2 ) F w> = 668 5 - 38 6 - 30 - 029-6 
Hence the efficiency of this Rankine cycle is 



629 6 
v 

Tins example will seive, incidentally, to show how unimpoitanL 

is the feed-pump teim. It i educes the amount of woik done by less 
than one pait in six hundied II we had left it out of account, and 
baken the heat-drop in full as the numeiatoi in icckomng the 
efficiency, the fig me obtained would hn\c been 02923 the 
cliff ei cnce is insignificant' 1 . 

A Cainot cycle with the same limits of tcmpeiatuic Mould (Ail 
81) have the efficiency 320 The difference between this and 292 
shows the loss which i esults in the Rankine cycle fiom not supplying 
all the heat to the best possible Ihcimoch namic ad^ antagc, namely 
at the top of the Icmpuatuic lange It amounts in this inslance 
to not quite 3] pei cent of the whole Heat -supply 

90. The Function G. In his steam tables Callcndar gives 
numeiical values of a iunction G, defined by the equation 

G =- T<f> - /, 

which applies to steam in any state, wet, div-satmalcd, supei- 
heabcd, or supei cooled Bv the help of this function the piocess oi 
calculating the hcat-diop ma\ be slightly shoitencd G has the 
impoiLanl propel Ly that it js con-slant dining a pioeess of e\ apoi.i- 
Lion or condensation al constant pressure Foi in any step of such a 

* Accordingly <i good approximation to tho c'flioioiic 1 } oi flic R.inkiao cyolo is 
obtained by leavniy out tho teim A (P a - P 2 )V tl , , in both iuimi.ia.loi and doiionu- 
natoi of tlio complete oxpiession jn Ait SS, and willing it simply 



in] THEORY OF THE STEAM-ENGINE 103 

process S/ = TS(/> and T is constant, consequently SG = Hence 
the value of G for a wet mixture at temper at me T and entropy (f>, 
such as the mixture at c (fig 20) resulting from adiabatic expansion, 
is the same as G 3 , the (tabulated) value of G for dry-saturated 
steam at the same pressure Therefore to find I , the total heat of 
the wet mixture, we have 



The heal -drop is then determined as before, by subtracting I c 
fioin the tolal liCtit before expansion. 

Takino l he same numerical example as m Art 89, T is 311 81, 
is 1 5020 and G h (for saturated steam at a pressure of 1 pound 
per square inch) is 2 01 by the tables This gives 

/ - 311 84 x 1 5G20 - 2 61 = 48-1 5, 

and the heat-drop from the dry-saturated state befoic expansion 
is 608 5 - 481 5 01 181 as before 

Oi we may obi am I lie heat-drop even more directly thus, when 
labulalc'd \alues of 6' aie available The relation 

/ = T<l> - G 

holds foi any slate ol I he substance Hence between any t\\o 
poiiilt> (b) and (c) on the same adiabatic line the heat-drop 

/>-/ = (T, - T e ] </, - (G b - G c ) 

In I lie picscnt example G b is the value of G lot saturated steam 
a! /'= ISO, which (by Ihc tables) is 51 10. (^ is equal to I he 
value lor sahualcd slcain al P 1 , \\hich is '2 (>1 The dillcience 
of tcmpciatuie r l\ Y' c u> 150 71 devices IIoucc I he heat-drop ib 

150 71 1 5<>20 -(''5110-2 01) = 181 0, 

uluch agrees with the rcsull louiid abo\e by less direct methods 

The use of Hie fund ion G in llns connection is only a matter ol 
convenience The procedure m Art 89 gives Ihe heat-drop readily 
ciiouuh, though not quite 1 so shoilly, without the help of G 1 ' 

* G (with its sign rovoiwed) is one of Lliroo functions to which Willaid Gibbs 
gave tho ruimo of "Thormodynaimo Potentials" see hit) ticmntijic Papers, vol r 
llo roprosotifcod llioni by tho symbols ^, x, anc l i" Of thoso, \f/ IH E-T<fi This 
fiiiiotion was callod by Holmhollz (ho " FIDO Enoigy", it is used in the 
theory oi solution and otlici applications of theirnodynaimos to chemistry, a 
subject outsido tlio scope oi this book Tho function x *& tho total heat I, namely 
E +APV, and is, as wo have soon, of paitioular impoi tance m the thermodynamics 
of engineering Tho lunotion f is IS-Ttp+APV or I-T(j>, hence 0= -f This 
function is useful m treating of tho equilibrium of different states oi "phases" of 
tho same substance One example of auch equilibrium occuis in wet steam, which 



104 THERMODYNAMICS [cir. 

91. Extension of the Rankme Cycle to Steam supplied in 
any State. In the Rankine cycle described in Ails 80-87 the 
steam Mas supplied to the cyhndci in the d^-satuiatcd stale But 
the teim Rankme cycle is equally applicable whatever be the con- 
dition ol the woikmg substance on admission, whether wet, dry- 
saturated, 01 superheated As regaids the action in the cyhndci, 
all thai is assumed is that the substance is admitted at a constant 
piessme P l9 is expanded adiabatically to a piessurc P z and is 
discharged at that pressuie, and that m the piocess thcie is no 
tiansfei of heat to 01 from the metal, noi airy othei iircvcrsiblc 
action. In these conditions the heat-diop 111 adiabatic expansion 
fiom P! to P 2 is the tlieimal equivalent of the nic&jbcL in fig 22 
(compare also Art 80) and theiefore nieasines the woik done in 
the cylinder, no mattei what the condition of the subt, lance on 
admission ni&y be. This applies to wet s Learn 01 superheated 
steam just as much as to diy-satuiatcd steam 

92. Rankme Cycle with Steam initially Wet A complclc 
Rankme cycle for steam that is \\et on admission to the cylmdei 
is shown on the entiopy-tempeiatine diagram by the figuie ab'c'c 
(fig 24<). The point b' ib placed so that the lalio ab' to ab 

is a nurture of two "phases, " liquid and vapoui Tlio tuucfciona \j, and j 01 - 
will bu nici i id to aso-in in Chap VII 

From the engineering point of view it may bo useful to point out llial tlicso 
functions liavo the following piojpeity Retailing to Ai( bO, Jig 15, \vo lia\o aotn 
tliaL vi ken any fluid expands aduibuliudly liom any stato a to any utliei wtato h, 
tho tliemial equivalent oi the aioa eabj, 01 A\ VdP, is tho hoat diop, l (l I tl , and 
that Iho aioa tnabn 01 A\PdV is the loas ol intcinal onoigVi ^,,-^1, yjnulniJy, 
if ab 111 tliat clmgiam repiesent an isothermal piocoas wo Jiavo hvo concaponding 
pioporfitiona, uith iega,id to the lunctions and \f> When any !Juid o\panda 
ibothet mally hotu any atato a to auj r ftate b, tho tlioimal oqnualcnt of tho aiua 
cubf, 01 *Jj ( dP, is G lt -<?/,, and that of tho aioa malm, or A\PdV, IB \jj (l - i/v, 

To piovo Llus, wo have by definition ^ =E -1 tf> Honco in an isothoimal procose, 

d^=dE-Td<l> 

Eut Td<[> it) tho lioat Lakon in, winch IN equal to the gain of mtoaiul onoigy ylus 
tho woik clone, OL Td>/> = dE-i APdV 



I 1 ' 
Tlioiofoio d^--APdV, and $, t ->[>,,= A PilV 

J 
Agam, wo have by definition G Tfi-I Ilonoo m an isothoinial prooosa, 

dG = Td<f>-dI 
But (by dofuution) I=E-\ APV, horn which 



Thoi of 01 G dO=-A VdP, and 6 tt -G b =A\' VdP 

J a 





in] THEORY OF THE STEAM-ENGINE 105 

is equal to q 1 the assumed dryncss on admission The line b'c' 
represents adiabatic expansion fiom P l to P a , c'e icprescnts 
condensation at P 2 , and ea re- 
presents as before the heating of a ^^^^^^^' 
I he condensed watci 
The totr,l heat bcfoic acha- /y <; 

/fV X 

batic expansion is I Wi + q-jL-^ or 
/ Si (1 g t ) L l and the heat 
supplied is the excess of this 
quantity above p lg 24 



The cntiopy <j> during adiabatic expansion is 

<j> Ul + frLJTi or ^-(1-fc) ,/?', 
The toldl heat aftci adiabatic expansion is 

I^-T^-fr or T^-G,. 

The lieat-diop is gol by subtiacting this hom the total Jieat 
bcfoic achabatic expansion. Or the heat-chop may be iound, as 
soon as (/> is calculated, by uiing the expulsion 

(Z\ - T 2 ) - (G, - GJ 

'J'he cllici(.it> A\lnch, as bcloie, is praclicall) equal to the heat- 
diop cln uk'd by Ihc heat suj)phed, is slightly kss than when the 
slL.iin is saluiattd bclou 1 cxpdnsion, the icason beiuy that the 
})H)[)(jition ol huit su})plicd at the upjjci hunt ol tcmpeiatmc is 
no\\ lathi'i less, because pail of the watei icmains uucoiivei ted 
iiilo sleani 

As .1 uiiUK'iieal example let g l be 0, and let Ihc lunils of pussuic 
be Hie same as in Ihc example ol Ait 89 Then the total heat 
pei Ib of I he mixtiue bcloie expansion, which is / s lL 1) is 

008 53 - I X 170 2 = 020 9. 
The heat supplied is 020 9 38 9 = 5S2 The entropy is 



The total heat after expansion I Si T z ((f) Sz c/>), 01 T 2 </> G 2 , 
is 452 4, the heat-chop is theicfoie 168 5, and the same liguie is 
obtained for it by the duect foimula ( r JL\ T 2 ) (G : G 2 ). 
Allowing foi the feed-pump term, the efficiency in the complete 
Rankine cycle is 289, as against 292 when there was no initial 
tvetness. 



106 THERMODYNAMICS [en. 

In piactice the steam supplied to an engine \voukl be MU only 
if theie weie condensation in the steam-pipe, such as would occ'in 
if it weie long or insuflicientl} coveied with non-conduclmg 
matenal, or if the boilei "pinned " Pinning is a defective bodci 
action which causes nnevapoiated watei to pass into the slram- 
pipe along with the vapoui. The above example will show llul i\ 
modeiate amount of wetness leduces the ideal eflicicncy onl> VCHY 
slightly, it has no moie than a small effect on the figinc 1 lor I lie 
Rankine cycle. But its piactical eflect in reducing Llic dlic'ii-ncy 
of an actual engine is much gieatei, because the picsc-ncc of walri 
m steam increases the exchanges of heat between it and the nu l.il 
of the cylinder, and consequently makes the real action dc-parl 
moie widely fiom the achabatic conditions which are assumed in 
the ideal opeiatioiib of the Rank me cycle. 

93- Rankine Cycle with Steam initially Superheated On 
the othei hand if .the steam be supciheated before it enleis llu- 
engine, the exchanges of heat between it and the metal aio mlucnl , 
the action becomes, moie neaily adiabatic, and the puioim.mcc 
of the leal engine appioaches moie closely the ideal of I he Rankine 
cycle Tins is the chief icason why supci healing nnpiovcs I he 
efficiency of a leal engine of the cylinder and piston typo In 
steam tin bines it is beneficial paitly foi the same icason and p,u Uy 
because it i educes mteinal faction in the uoikmg fluid by kec pin" 
it dner than it would othenuse be during its expansion I In ougli UK- 
successn e imgs of blacks Supeiheatmg i^ now very gcnt-rairv cm- 
ployed in steam engmeeimg It is theiefore impoi taut to consult i in 
some detail the Rankine cjcle foi steam thai is initially supei hc.ilcd 
In the entropy-tempeiatuie diagiom (fig. 25) the line W iqiic- 
sents the piocess of snpei heating sLcam that was flry-saliii.il cd nl 
b. Dining this process its entiopy and its tcmpcratme bolh m- 
crease, and when the pressuie and tempeiaime at any sta<>(> m 
the supeiheatmg aie known the coriesponding cnliopy is ft.inul 
fiom the tables relating to supeiheatccl steam It we assume llial 
the picture dining superheating is constant, and eq.ml lo llu 
boilei piessuie, the line bb' is an extension, mlo the ,<., ,,r 
supeiheat, of the constant-picssure line ab Duiino H K - p.oc.ss 
of supeiheatmg the .team takes in a supplemciitaiy q.muhly of 
heat equal to the aiea undei the cuive bb', measmcd dmvn lo (he- 
base line, namely nbb'n'. This quantity of heat may also be fonnd 
horn the tables, being equal to the excess of the tolal heat /, 



Ill] 



THEORY OF THE STEAM-ENGINE 



107 





over thai of saturated steam of the same picssuie Callendai's 

tables give values of the 

total heat of superheated 

steam, as well as its en- 
tropy, iw a wide lange of 

picssines and tcmpe-iatuies 

Dining the subsequent pio- 

ccss ot adiabatic expansion 

b'c' the steam loses snpei- 

heat, and il the process is 

camed so fai lhat the adia- 

balic line tlnoiigh // ciosses 

the boundary curve, it be- 
comes sa Una led and then 

\vel, and the final condition 
is lhat of a wet mixline at 
c' The tolal heal ol this 
wel imxlnic is found b\ the 
method aheady dcscnbcd 

The uoik done in the 
Uankmc cycle as a vhok. 
is Ihe aiea eabb'c', and Ihe 
heal taken in is the aiea i 

[tithh'n 1 Bolh the.sc quan- 
tities ,ne ie\i(lily calculated 
without the help of the chaplain 
cycle we h,i\e onlj l<> calculate the heat -chop dining adiabatic 
expansion, namely, I b > - /,', and subtiact fiom that the small 
In in \\liifh is Ihe Iheunal eqm\ alent of the woik clone m the teeel- 
pump, namely, / (1\ - P,} /',, The heat supplied is 

As a niiinc'iic.il example we. may again take Pi = ISO andP, = 1, 
and assume lhat supciheating is earned so fai as to imse the tem- 
pei-aluu of the sleam to 400 C , which is a hunt veiy laiely ex- 
ceeded in piaelice As a uile the tempciatmc aftei supeiheating 
is eonsuleiably lowei than this. With these data the steam tables 
show that Ihe total heat of the supciheated steam is 7SO 8 and 
Us enliopy is 1 7033 The heat-supply is 780 8 - 38 9 = 741 9. 
Al'lcr achabatic expansion the steam is wet, and its total heat, 
which is /,, - 2' 2 tf st - 0) 01 T^ - G 2 , is 547 2 The adiabatic 
heat-drop is therefore 233 G 



Er<t ropy 
Fin 25 



n n' 



To luul the \\oik done in the 



108 THERMODYNAMICS [OH. 

Or we can find the heat-chop very directly by help of Callendai's 
values of G By Art. 90 it is (T/ - T 2 ) - (G/ - G 2 ), \vheie 1\' 
and GI icfer to the initial state aftei supeihcating. This gives 
361 3 X 1 7033 - (406 09 - 2 61) which again is 233 6 

If we deduct the small feed-pump term (0 3) the efficiency of 
the cycle as a whole is 233 3/7-11 9 - 314 This is lathei better 
than the figure foi satmated steam (0 292) because a poition of 
the heat is now supplied al a highci tempciatmc. Even with the 
extreme amount of superheating, howevei, which is assumed in 
this example, the mam pait of the heat is still supplied at the tem- 
peiatuie of saturation, and therefoie there is no gieat gam in 
theoictical efficiency as expiesscd by the ideal Jiguie for the 
Rankme cycle. The piactical advantage of superheating is much 
more consideiablc, foi leasons which have already been indicated, 
than might be expected from this companson 'of the two ideal 
cycles. 

In the adiabatic expansion of supeiheated steam a state of 
saturation is i cached when the pressuie falls to such a value that 
S foi satmated steam at that pressuie is equal to the entiop}' 
duimg expansion In the numerical example the entiopy dining 
expansion is 1 7633, and the tablet, show that this corresponds to 
satuiation at a piessme of 136 pounds per squaie inch. Any 
fuithei expansion produces wetness, or else supei saturation 

If it be desned to tiace the changes of volume dining the adia- 
batic expansion of supeiheated steam, the initial \olume (conc- 
spondmg to the assigned piessme and temperature) -\\ill be found 
in one of the steam tables (see Appendix III) The formula 
P (V by 3 = constant then applies, down to the picssuie at 
which the steam becomes satmated In this foi mul a, as was 
explained in Ait. 78, b is the volume of watei at C , namely, 
0160 cubic feet; a teim so small that it can usually be left out 
Dining furthei expansion, when the steam has become wet, the 
volume at any stage may be deteimined (as in Ail. 78) by fust 
calculating the diyness q. 

Moie cluectly, and veiy exactly, the volume of a wet mixture 
is found (without calculating q) by Callendar's foimnla (Art. 211) 

l s Kl 

whcie K is the minimum specific heat of watei, namely 9967, and 
t is the tempeiatuie measuied fiom C Since K is veiy nearly 



in] THEORY OF THE STEAM-ENGINE 109 

unity it makes no sensible dilfeience in this formula to write t for 
Kt and we therefore have 

17-17 ^ q ~ ^ 
r <1 ~ y s r f > 
J. s b 

as a convenient means of finding the volume of a wet mixture at any 
tempciaturc t, when the total heat (7 a ) of the mixture is known. 

To exemplify these methods of finding the volume we may take 
the same case as before, namely the expansion of steam at P = ISO 
pounds per sq. inch and tcmpciatme 100, down to a final 
picssme of 1 pound pei sq. inch By the tables the initial volume 
is 3 9605 cub ft We thcicfoie have, dining the fiist pait of the 
expansion, while the steam slill lelains some superheat, 

1 3 log (V - 016) - 1 3 log 3 9145 + log ISO - log P. 

Tins applies down to P 13 62, the pressure at which saturation 
is i cached Applying it to that pressure we find V = 28 7, which 
apices as il should do \\ilh the volume gn en in the table foi satur- 
aled steam. Assuming that in the subsequent pait of the expan- 
sion the si cam is in equilibrium, it will be wet, and its volume V ^ 
is (ound al am stage by (nsl limling / at that stage and applying 
[he mo'lhod gi\ui above In Ihc final condition, when P 1 and 
/ 51? 2 I lit- volume so calculated is 295 2 cub ft and the div- 
lU ss is SS(j 

94 Reversibility of the Rankme Cycle \Vhatc\ ci the 
initial slalc be, Mhclhei diy-salmalcd, wet, 01 superheated, the 
mtiinal achon ol Ihc woiking siibslancc in Ihc Rankme cycle is 
leviisibli An ideal engine [Kifomung a Rankine cvcle may be 
legaided .is a slnclly U'\ r i isihk engine taking in heat at various 
Icmpiiatuies (Ait 1-3), and consequently extracting the gieatest 
j)f)ssil)le aniounl ol woik out of c.ich clement of the heat supplied, 
having legaul to I hi tcmpetaluic al which the clement of heat was 
supplied In the healing of Ihc feed-water a pait of the heat- 
supply is taken at (cmpciatm.es ranging fiom T, to 1\ But am 
element ol beat, Liken in at a tempeiatinc T, acts as efficiently as 
it would do in a Carnot cycle the efficiency of conversion of that 

T T 
element is equal to ~ fT - '. Consequently the general efficiency 

of an ideal engine working on the Rankme cycle is the highest 
possible ediciency that is compatible with the condition that the 
substance is to be completely condensed at the lower limit of 
tcmperatuic and returned to the borlci by a separate pump, instead 



110 THERMODYNAMICS I' 1 "- 

of having its cycle completed by adiabatic compression as in UK- 
engine of CainoL 

In othcL- words, the woik which the steam does in the c\ hndri 
of an ideal Rankinc engine is Lhe greatest amount of uoik thai 
can conceivably be done by Lhc steam in passing Ihiough am 
engine., haMng regaid lo the tempera tin c at winch the \M>iknui 
substance has taken in its heat, and to the icnrpeialuie a I \\hich 
it icjccLs heal during its complete condensation bcloie bung i<- 
tinned to the boilci But we know that this uoik is nu.isuml b\ 
the adiabatic heat-drop Consequently I lit- adiabalic heal-dn>p 
measures the greatest conceivable pei lonnanre ol UK- sh.un 111 
passing through any engine wlrur Ihe conditions ol supph .uul 
of condensation aic assigned. 

Whatevei therefore be the nalnic of the 1 engine., the ndiaUitu 
heat-drop scivcs as an ideal standard with which lo eompait UK 
actual performance Thus a steam turbine, equally \vilh an < nuim 
of the cylinder and piston type, cannot exceed, and IK eissaiiK 
falls short of, the ideal performance- as mensuu-d by lli.il ln.il- 
drop In the design of steam turbines the ealeulah d valiu <>l I IK 
adiabatic heat-drop, after making a deduction which is dcl( inniu d 
b} T experience with similar machines, accoichnglv toiins UK l),i i- 
on which the dct-iguci estimates the pcifounance lo ht evpcch d 

In any engine the ratio of the actual amount ol uoik di>n< 
per Ib. of steam to the amount that would be doiu in UK id al 
Rankme engine under corresponding condihons ol snppK .ind 
exhaust, is called the Efficiency Ratio'} . Tests of^ood engiiK s S|K.\\ 
that in favomablc cases the actual ]K i foimanee is about 70 pi i 
cent, of the Rankme ideal About 70 pel cent, ol Ihe adml>alu 
heat-drop is actually converted mlo woik. 

95. Conditions of High Efficiency. To seeiin high I'llieu nc\ 
in the conversion of heal rnLo woik there are obviously lwos pai.id 
conditions to be aimed at (1) that tlieie shall be n lai^i- IMJI!- 
drop relatrvcly to the heat of formation of Llu sUam in nlliu 
words a high value for the ideal elTicicncy, (a) lh,d Iheie shall Ix 



* To facilitate such ostmmtos tables aio puLhsIiod K'vinfj; (ho hoii(.ilni|) uiuli-i 
a wide range of itulml conditions as lo ])ICSHUIO and Hupoihoai, mid dual CIHK|I(II.I) 
as to pressure of condensation Those aio founded on Onlh'iidai'H Kloaiu Tnl.lt << 
See Heat Dtop Tables, JI Moss (Edwt-rd Arnold, 1917) Tho Htuilciii will liml il n. 
useful exercine to compare tlio values thoro given with tho honl, ilum an calcMilfifnl 
by the methods of Arta 89, 90 .ind 93 

t See Eeporfc of a Committee on tho Thoimal Eflu-ifiioy of ,St<vuu 
Mm Pwc In<tt Civ,Eng,\(A oxxxrv, 1898 



Ill] 



THEORY OF THE STEAM-ENGINE 



111 



a large Efficiency Ratio The second condition depends on piactical 
features of design with which we aie not at piesent concerned 
But as regards the ideal efficiency it is impoitant to notice that 
while some advantage is obtained by increasing the admission 
pressure, a fai gieatei advantage is obtained by lowering the ex- 
hausL piessure. 

That this is so will be clear fiom the following tabulated result^ 
which relalc to saturated steam The fiist table shows how the 
heat-diop and the efficiency of Lhe Rankme cycle aie affected by 
taking diffcienL initial pressmcs, langiug fiom 100 to 300 pounds 
per squaie inch, but with thii"same piessure of exhaust throughout. 

Rankinc Cycle f 01 S titillated Steam Effect oj vaujmg the 
Initial Piessiuc. 



Initial Heat-chop, to 
pressure a final pros- 
( pounds per wuro of 1 Ih 
squaio inch, per squnio 
absolute) inch 


Woik done 
per Ih of 
steam allowing 
for \\oik spent 
in feed pump 


Hoat 
supplied Ehcipnc\ of 
per Ih the Rankmo 
oi steam 03^0 lo 


300 


(11) CillollCS) 

2020 


(Ib caionos) 
201 5 


(Ib calorie^) 
034 8 


0317 


280 


1906 


199 I 


0342 


o m 


200 


1970 


1 96 6 


033 4 


0310 


240 


1942 


1938 


032 


300 


220 


191 1 


1908 


031 7 


302 


200 


JS7 7 


1874 


030 7 


0297 


ISO 


1810 


1837 


029 () 


0292 


]()() 


1798 


1795 


0283 


0280 


1 10 


17f>0 


174 8 


026 8 


279 


120 


11)95 


1<>93 


0251 


0271 


100 


102 9 


102 8 


023 


0201 



Kumi llusc icxulls jl will be appaienl that \\hcn (he admission 
pu'ssuic is high \ r ciy little jinpiovemcnt in tlic efficiency is biought 
about by even Ji laigT mcicasc 1 of pic-ssuic Tlic twenty-pound use, 
fo i example, 1'iom 2SO Lo 300 augments the efficiency bv onl} r one 
pt-i cent 

On Ihe oilier lumd it is oi gieal tidvanLigc to have what eiigmeeis 
call i "high vacuum" that is to say to make the piessuie of con- 
densation as low as possible. If a high vacuum can be maintained 
and effectively utilized we obtain fiom the steam the work which 
il is capable of doing under conditions of low picssme but of vciy 
large volume in the last stages of the expansion The following 
table illustrates the gam in hcat-diop and m efficiency that 
results, in the Rankme cycle, from reducing the lower limit of 



112 



THERMODYNAMICS 



[cir. 



pressure. In this example the admission piessure P l is assumed 
to be 180 pounds per squaie inch, and only P 2 is alteied 



RatiLme Cycle fo> Salivated Steam 
Final 



Effect of varying the 



Final 
pressuie 
(pounds pci 
squaie inch, 
absolute) 


Heat-drop Work done pei 
fiom an Ib of steam 
imtialpiessure allowing toi 
of 180 Ibs work spent m 
pei square inch feed pump 


Ho.it 
bupphed 
pei Ib 

oi steam 


Efficiency of 
Llie Rankmu 
cycle 




(Ib calories) 


(Ib calones) 


(Ib. caloaes) 




4 


144-5 


1442 


bOl 1 


0240 


3 


153 1 


152 8 


1 607 5 


0251 


2 


1649 


1646 


610-1 


0267 


15 


1730 


J72-7 


621 8 


0278 


1 


184-0 


183-7 


0290 


0-292 


0-5 


2018 


2015 


6120 


0-314 



The last fig me corresponds Lo a vacuum of neaily 29 inches ol 
meicmy with the barometei at 30 inches. 

To secure in a teal engine the full benefit of a high vacuum lheslc.ua 
must continue Lo do useful woik in expanding down to the piessuie 
at which condensation is to take place In engines of the cylmdei 
and piston type this is impiacticablc loi two i caserns the volume ol 
the steam becomes excessive, and the mechanical I'ucLion oi the 
piston against the cylmdei becomes i datively so gi eat as to dbsoi b 
all the woik done m the final stages. But with the steam tmbine. 
these considerations do not apply, theie is then nothing lo piovcnl 
the steam fiom continuing to do useful woik as it expands ughl 
down to the piess>me of the condcnsci, and special p.uns ;uc 
accoidmgly taken to maintain a good vacuum in the condenser 
of a steam tuibme It is laigely (oi tins icason th.it good sU.un 
tin bines achieve in piacticc a gicatei cllicicncy lhan even I he besl 
engines of the cylindci and piston lypc (See AiL. 150 ) 

96. Effect of Incomplete Expansion. When steam is ic- 
leased fiom the cylinder at a picssiuc substantially higher than 
the piessuie m the condensei its expansion is saiel lo be incom- 
plete. The effect is to lose available Avoik rcpicscnted by the toe 
that is cut off the piessmc^ olume diagiam, as in fig 20, and lo 
make a coiiesponclmg reduction in the clliciency Release takes 
place at c and the piessure falls to/ while Ihc piston is stationaiy. 

To exhibit incomplete expansion on Ihc entropy- tcnipern hue 
diagram, imagine that instead of letting part of the steam escape 



JIIJ 



THEORY OF THE STEAM-ENGINE 



113 



Fig 20 



irom the cylinder by opening the exhaust- vah e, -\\e pioducc the 

same effect within the cylinder itself 

(as might be done in the engine of Art 

SI) by applying a icccivcr of heat 

which will bung the picssuie clown to 

the lower limit P 2 by causing pait of 

the contents lo condense bcfoie the 

piston begins its ictmn stioke The 

piston being stationai}', the volume of 

the working substance does not altci 

dining this pioccss If we imagine the icceivei of heat to have 

a Lempcratnvc which falls progressively fiom that of the steam 

al 6- to lhc final tcmpciatuie (T 2 ) at/, this icmoval of heat takes 

place reveisibls The work done by the steam is not affected by 

substilulmu this icvcisible piocess for the action of the condenser, 

because Ihc pussinc in Ihe cyhndci is in no way altcied by the 

sul)stitulion, but we aie now able to diaw a cm\c that will 

iipusinl tin puxiss on Hie cnliop\-tcmpciatuic dingiam 

This is dom in ML> 27, \\hcic the cm \e cf icpiescnts the con- 
(U nsalion ol pai 1 ol 
I he sli'.im al constant 
\ olmm , \\ Ink I IK- 
piston is al Hsl bc- 
(01 < hi yummy il s 1 1 1 - 

I III II si I ()1\C Till I'OII- 

slanl \ oluini in I his 

pioci'ss is lo hi- ii ck- 

onc d pet II) ol steam 

il is lhc \oluinr ol 

lhc c\ Imdi i (lixidul by lhc qnanliU of lluid mil m olhri woids 

il is llu \ohmii pi i Ib ol Ihr \\ct slcam al c C'all lh<il volume l\ 

Tin n al any li'\ 1 1 of li'inpi'ialmi' such as gill, a poml i on the con- 

slaiil-volumi' cuive which lepicsenls Ihe pioccss is loiind by 

lakmg gl ^ p o 




where V n is lhc volume of 1 Ib of satmalcd steam at lhat tem- 
peiature. The area of the liguic within the shaded lines rcpicsents 
Ihe Ihcimal eqmvalenl of the woik clone in the complete cycle 
The coiner cut off by the curve cf shows what is lost by incom- 
plete expansion as compared with the work done in a Rankine 




114 THERMODYNAMICS . [en. 

cycle. In the example sketched in fig 27 the initial piessurc (at b) 
is ISO pounds pci squaie inch, and the steam is icJeascd after 
adiabalic expansion to 15 pounds pci seniaie inch. 

97. Ideal Engine working with No Expansion. If acha- 
batic expansion weie entnely absent, and steam weie admittcel 
at Pj dining the whole of the foiwaid stioke of the piston, and 
dischaiged at P 2 elm ing the backward stioke, the cntiopy-tcm- 
peiatuie diagram would take the form shown in fig 28, wheie bif 
is a constant-volume 
line lepiesentmg the 
fait of piessuie fiom 
P! to P 2 This coi i e- 
sponds, in the ideal 
cycle, to the action 
of pinmtive steam- 
engines such as New- 
comen's, bcfoie Watt 
introduced the piac- 

tice of cutting off the supply of steam at an eaily stage in the stioke 
and allowing the icma.in.dei of the stioke to be pcifoimed by 
expansion under falling picssine Points in the cm ve bij aie lonnel 
as in Ait. 96 

In this case the woik done in the cjlmdci, pci Ib of steam, is 
(P x P 2 ) V^ The net amount of work done, allowing ioi the 
feed-pump, is (P x - P 2 ) (F % - F Wj ), and the Lhcimal equivalent 
of this quantity should be equal to the aiea within the shaded 
lines, of the entiopy-tempeiatme chagiam As a mimciic.il example, 
assume P t to be ISO pounds pci sq inch, and P l?o be 1 pound 
per sq. inch, which aie the prcssuics foi which fig 28 has been 
diawn. Then the woik done per Ib is 179 x 144 x (2 562 010) 
loot-pounds or 47 17 calories, in companion with the 183 7 caloncs 
of the Rankme cycle foi the same initial and final pi assures (see the 
table in Ait 95) The heat supplied per Ib is 020 G caloncs; the effi- 
ciency oi the ideal engine without expansion is fhcicfoic only 0749 
The efficiency of actual pumitive engines woikmg without ex- 
pansion was much less than this, not only because the pressure 
\\as less, but because at every stioke a laige part of the steam 
entcimg the cylmdei became at once condensed upon the walls, 
and consequently the volume of steam taken fiom the boilci was 
gieatei than the volume swept thiongh by the piston 



m] THEORY OF THE STEAM-ENGINE 115 

98. Clapeyron's Equation. This name is given to an im- 
poitant relation between the latent heal of steam 01 any other 
vapour, the change of volume which it undeigoes in being vapor- 
ized, and the late at which the saturation piessme varies with the 
tcmpciatiirc To establish it we may i evert to the ideally perfect 
steam-engine of Ail 81, m which Carnot's cycle is followed with 
a liquid and vapom for woikmg substance We saw that this gave 
an mdicaloi diagiam (fig 10) \\ith two lines of umfoim pressure 
(isolhermals) connected by two adiabatic cui\es. The heat taken 
m was L pei unit of woikmg substance, and since the engine 
\\as icveisible its efficiency was (I\ T)/'l\, fiom which it 
followed that the woik done, 01 the area of the diagiam, was 
L (7\ T 2 )jT l This is m thermal units to reduce it to units 
of woik we multiply by J Now suppose that the engine \\oiks 
between Iwo tempciatuies which dil'fei by only a veiy small 
amount We may call the tempciatuies T and T ST, ST. 1 being 
the small mteival thiough which the engine woiks The above 
cxpicssion ioi the ^solk done becomes 

JLS7' 
T 

The indicatoi diagiam is now a long nanoM stnp (ilg 20) Its 
length cib is V ^ - l\ u , V \ being Ihc 
volume oi unit mass ol the \apom and 
/", Ihc \ olimu of unil mass ol the liquid 
Its height is Bl\ \\heicSP is the dillu- - 
cnce between the picssuie in ab and ^ 
Ih.il m cd In olhcr woids, since Ihc a 
\ai)oiu is sal mated m cd as well as in 
ab, S/ > is the dil'fcicnce in the picssurc 
oi scihualcd \apoui due lo the dillcr- 
cnce m Lcmpcratiuc 87" When 8P is lfi 

made vm small, the aiea of the duigiam becomes moic and moie 
ncaily equal lo Ihc piocluct of the lenglh by the height, namely 
S/ J (F, - F w ) Tins is equal to the woik done, whence 



This equation is only appioxnnate when the interval SP (or ST) 
is a small finite mtcival In Lhc limit, when the mteival is made 
indefinitely small, it becomes exact, and may then be written 



F _ V --- \ 1 

' " w T (dPJs 9 



82 



116 THERMODYNAMICS |cu. 

dT 



where (jn) means thciatcal which I ho lempeiaUne of salmaUd 

\-i J s 

vapour changes iclalivcly to the piessure 111 other uoids il is the 
slope of the satuiation cuive of lempciahne and pussme This 
is Clapeyron's Equation II may be applied t<> hnd I lie \olinne <>l 
a satuiated vapom, at any temperature, whi'ii the \olunu ol Hie 
liquid, the latent heat, and Lhc lale oi' change ol tempt latuie \\ilh 
presume along the satniation cuivc are known. 

The values of pi ess nre, volume and LiU'nl lu;il gi\en in steam 
tables m iclation to tempera Lure, must, if Liu- tables aie .icciu.ili , 
be such as will satisf}'' this equation Take, loi t \ani pic, sh niu nl 
150 C. Callcndai gnes 67-313 pounds per ,s([iiai< mcli loi Hie 
satuiation piessme at 119 C. and 71 025 .il 151" (' Tin- i.ili at 
which P is changing pei degice al 150 may be laken .is lull (In 
difference, or 1856 x 141 pounds per sqnait loot /', is 0017 
cubic feet, and the latent heat is 50050 ealmus He net 1>\ 
Clapeyion's foimula we should find, foi llu s.iluiMlum \oluine in 
cubic feet, 

v Annr , 1400 x 50G 5(J 

V. = 017 -(- - (, "S!) 

423 1 x 1 850 X 1 1 1. 

This agices with the labuLiltd value ((> L',s<)5) 
We might ha\e obtained Ihe ClajKyion equ.ilion l>\ 
that the entiopj -tempera I me diani'.un eoiiespoiidmn 
dicator diagram of fig 20 is a loni- naiiow ship, ^ 
<j> s - <{> w is L(T and heiyhL is BT Us art a is I be 1 1,, im.,1 , () ,,,\ .,[< ,,| 
of the woik done, henec SP (F, - rj JL^Tjr, as l, l,. l( 

In the vaporization of a liquid /" is uinln than / (ll , ,md 1,,-al 
is taken in, hence by Clapeyron's EquaUon (^ ,s ,,s,l n -. u I,,, h 
means that mcicasmn Ihe jnessme uuses ||,,. |,,,,| m ,, ,,,, u ,, ( 
the change of volume V,-V W is known fo, , 1MV s,,bsl.,, l( , ||,,. 
equation may evidently be used lo lind ll,e ..nu.iml |, x ul, l( |, ,|,, 
boiling point is uused pei nniL mnease oi pnssn.e 

99 Application of Clapeyron's Equation to other Changs 
m Physical State The leasonmo | )y wlm . h Mlls ( n ' 

aiiived at was general a applies lo any .eve.sible | 1!U ,, M , 
he state of aggregation of any subslanee, lf> ,, ,,,,,. } [nm 
ohdtohquidas wellasto the change Jrom l.qu.d , ^ Th| 
engine whose mdieatoi diagiam was skdehed in |, 2 ,' I1HIV ,, . 
anything for working substance, and ,he IM ,H,,M m,l , , J 




in] THEORY OF THE STEAM-ENGINE 117 

fhst opeialion, during winch heat is taken in, may be drawn to 
icpiesent the change of volume coirespondmg to any change of 
state In the example already dealt with, the change was fiom 
liquid to vapom But we might begin with a solid substance pre- 
viously laisecl to the tempeiature at which it begins to melt (under 
a given picssure), and diaw the line to icpiesent the change of 
volume that occuis in melting, while the pressure remains con- 
stant. The substance does external woik against that pressure if 
it expands in melting, 01 has woik spent upon it if (like ice) it con- 
tiacls in melting The steps of the aigument aie not affected, and 
hence the equation may be wntten thus, with icference to any 
such tiansfoimation of state, 

V" V = 
T dP' 

where V is the volume of the substance (pci unit of mass) in the 
fiist state, V" its volume in the second state, A is the heat absorbed 
in Ihc liansfoimahon, and dT/dP is the late at which the tempera- 
line of the tiansfoimation (say the melting point 01 boiling point) 
is uUeiccl by alteimo the piessmc. 

If a solid body expands in melting V" is gieatei than V and 
(since the latent heat A is positn e) it follow s that dTfdP is positive 
in ollici woids the melting point is laised by applying pi ensure 

On the othoi liand if the body contiacts m meltino V" - V is 
m-ivitive and dP/dT is motive in othci voids the melting point 
is Imveied by applying piessine This is the case \\ilh ice. Fiom 
[lie known amount In which ice contiacts \\hcn it melts, James 
Thomson (in 1819) applied this method of leasonmg to pi edict 
thai the mcltiivj> point of ice would be lo\\eied by about 0074 C. 
j 01 each atmospheic ot picssuie, <md the lesult was afteiwaids 
veil fled expciimcntally by his biothei, Loid Kelun*. 

The lowei of the two fixed points used in giaduatmg a thermo- 
inolei (Ail 15) is the tempeiatme at which ice melts under a 
picssuie of one ntrnosphcic If this pressme weie lemoved, as 
it miffht be by putting the ice in a ]Ri exhausted of an by means of 

* See Kolvm's Mathematical and Physical Papers vol i, p 156 and p 165. The 
numerical result stated m the text is obtained as follows -A P^dof -water 
changes its volume in f tewing h om 0100 to 0174 cub ft , and gl ves out SO calories 
< 1T 00014x273 



_ 
dP~ 80x1400 

and if SP be one, atmonpheie or 2160 pounds per sq ft , ST is 2160 x OOO00341 
O j 0074 C 



118 THERMODYNAMICS \CLI. 

an air-pump -the tempciature of melting would be raised. The 
waters apom given off at the melting point has a prcssiue of only 
09 pounds pei squaie inch, and consequently if 110 air wcic 
present, and if the only picssuic wcic LhaL of its own vapom, 
ice would melt at appioximalcly 0071, C., foi the prelim- would 
be reduced by neaily one atmospheic The icmpeialme al 
which ice melts unclei these conditions is called Llie Tt tple PouiL 
because (in the absence of air) walci-slufl can c:?cisL iL I ha I par- 
ticulai tempeiatuie in three states, as ice, us water, and as vapoui, 
m contact with one anothci and in eqmhbunm. 

100. Entropy-Temperature Chart of the Properties of 
Steam. Besides serving to illustiatc the operations of ideal 
engines, a diagram in which the cooidmatcs aic the en I ropy <nul 
the tempeiatuie may be used as a gcncial chart foi exhibit mo 
graphically the piopeities of steam 01 of any oLlier fluid The 
student will find it instinctive lo chaw foi himself < clisul lot 
steam, on section paper, to a scale laifto cnoug]! [ O i le.isoiubly 
accuiate measuiement 

The general character of such a, cliail will be nppnicnl liom 
fig. 30 It includes the boundary cmvcs ahcddv tlcsc-i ilxd, uluch 
icpiesent the i elation of enliopy to Icinpcratniciii snl in .ilt'd ^l( .nil 
and in watei at the same tcmpciahiicaiul picssmr. 15i-f wccn llu-st- 
is the \\et legion, wheie Ihc subst.iuoe can CMS I in ctpiilibiium 
only as a nnxtuie of liquid and vapom Ucyoiul i b( sic am 
boundary, to the noht, js [he icgion of supc i healed vnpom 

Now let a system ofLtncs of Con^ftuil J'lCbwuc l>e din vvn I^.ich 
of these shows the lelation of c/> lo T while I lie snJ>sl,mer cliann<s 
its state in the mannei of tliL nnaj>mary e\.|)cun iciil ol Ail r>(> 
Staitmg fiomthe extieme Icll, ahnc ol conslaiil j^rc'ssui e I'm ^.ilei 
is piactically indistinguishable liom (he boundary C'ni\ f e, sdicllv 
it lies a httle to the left of Lhatcuive, le.ic'hinn it only wlun (lie 
tempeiature is such that steam begins lo foi in. TIic'ii il eiosst s Mu 
wet legion as a hoiizontal sliai^hl IHK-, T lx m^ <-onslnnl diiiinn 
the conveision of the siibsLance from IKJIIK! mU> vapour Allei 
reaching the steam boundaiy (he line ol 1 couslaiil pit'ssuie uses 
rapidly during the pioccss oL' .supcihcaliiiji. Tn Hie fi-imc, live 
lepresentahve lines of conslanl pressure- are (hawn, n.nnelv I hose 
foi P = 2, 20, SO, 200 and 500 pounds IKV sciuarc melt. When a 
sufficient number of such lines aie dia\vn it is easy, by f-iMphie 
mteipolation, to maik on Ihc clmil Iho position ol a })oiul 



Ill] 



THEORY OF THE STEAM-ENGINE 



119 



coi responding to any assigned condition of the substance as to 
picssmc and tcmpeiatnre, and to tiace, by measurement instead of 
by calculation, the changes which ensue dining adiabatic expansion. 
The convenience of the chait foi such purposes is increased by 
including a system of Lines of Constant Total Heat. Examples of 
these lines aie shown in Pig 30, for each intei val of 25 calones fiom 



1*800 



400 



350 



300 




075 



1 50 



I \ . I 

10 I 25 

Enuoov 
Vi<f '!(> Kriliopy Li mpoiiiluio (Jli.u I Im W.iltu ,iud ( Sl 



20 



1 25 



/ = GOO to / =. SOO c.iloncs The} aie si)ccicilly useful in I he icgion 
of supeihodl, llic}. m,i\ ho\ve\ei be diawn in the \\eL 10151011 also 
As nil example Lhe line foi / = 050 is conLmiic-d inLo Lhc \\cL 
legion, iL undeitiocs a sluii[) eliange of dueeLion in crossing Lhc 
slciim boundaiy Each of lliesc lines icpiescnts whaL occius in a 
Ihrollhng process The lines of constant toUil heat tend, at Hie 
extieme righl, to become neaily straight lines of constant tempera- 
ture this is because the vapoui behaves more neaily like a pericct 



120 THERMODYNAMICS [en 

gas the moie the pressure is i educed In a pcifecl gas, as we saw 
in Ait. 19, T is constant when the expansion i^ of such a nature 
as to keep / constant. 

Another useful addition ih a set of Lines of Con&tanl Dii/ness in 
the wet ieg ion. These aie diaun in the nguie foi <y = 1, 2, 3, 
4, 5, 6, 7, 8, and 9 They divide each honzontal width 
between the two boundaiy curves urlo equal paits (see Ait SI) 
Lines of Constant Volume may also be drawn in the mannci aheadv 
descubed (Ait 96) 

With the aid of such a chait one may find, foi example, by 
di awing the appiopnatc adiabatic (yeitical) line, that steam wilh 
an initial piessine of 200 pounds pci squaie inch, superheated lo 
400 C., becomes satin atcd when its picssmc falls, by adinbal it- 
expansion, to 16 pounds. Continued into the wet legion the adia- 
batic cuts the constant chyness line q = 9 at 50 C , showing lhal 
theie is 10 per cent of walei present when the piessmc has fallen 
to 1 8 pounds Theheat-diop may be mleiicd, but for its mcasuic- 
ment a bcttci foim of chait is one which will be dcscnbcd in the 
next Article 

By chawing a \ertical line to lepresent the adiabatic expansion 
of a mixtuie of steam and watci, it is easy to tiace Ihc changes 
that occur in the piopoition of walcr to steam In the icoion ol 
oidmary woikmg piessuies the line foi g = 5 is neaily veitieal. 
Hence if theie is about 50 pci cent of water picscnt at the begin- 
ning oi achabatjc expansion, neaily the same pcicentage will be 
found as the expansion goes on When the steam is much wcltci 
than this to begin with, adiabatic expansion makes it clnei. 

In fig. 30 a con]cctuial cmve has been added (shown by i 
bioken line) connecting the watci and steam boundaiy cmvcs in 
the legion ol high piessme, whcie, at present, Ihcic aic no dal.i 
foi a pi ease dctcimination of the cutiopy Tins bioken line is 
simply a smoolh cnne foiming a eontmuahon of each boundaiy 
cuive, and diawn so thai it (ouches Ihc isotheimal foi 305 C , lhal 
being the cntical U'lnpcialnrc foi walei It is a I the culic-al lem- 
peratuic thai the distmclion between <, and ro disappcais The- 
honzontal mlciccpt between the water and steam boundaiy 
cuives, which coiicsponds lo Ihc (akmg in of latent heat, Iheic 
vanishes the culical point is Iheiefoic at the summit of I ho T<j> 
cinvc At sufficiently high piessuies Ihc lines oi constant piessmc 
would pass, in the foim of continuous cmvcs, clear of the lounded 
top, fiom the legion of walei to lhal of snpei healed steam. 



in] THEORY OF THE STEAM-ENGINE 121 

The water boundaiy cmve is concave on the left for tempeia- 
tiues below 250 C., because the use of entropy pei clegiee, which is 
rr/T, wheie a it, the specific heat of watei, becomes less as T in- 
creases, a being ncaily constant at low temperatures But afc 
highei tcmperatiu cs the specific heat of watei increases so fast as 
to make a/T mciease with using lempeiatuie the curve accoid- 
ingly bends ovci to the light a-, it appioachcs the cntical point. 

In the next chapter we shall have occasion to icfei to examples 
of cntiopy-tcmpcialme cluuts foi other fluids In one of these 
caibonic acid the. icgion which is practically impoitant, in con- 
nection with icfiigciating piocesses, includes the lounded top 
whose summit is the critical point In that chagiam the lines of 
constant piessme in the liquid aic cleailj distinguishable fiom the 
bounddiy curve ol the liquid slate 

101 Molher's Chart of Entropy and Total Heat While 
the cntiopy-tcmpciatiue diagiam is invaluable as a means of 
exhibiting thcnnoclMiamic cycles and as a help towaids undci- 
standmg them, another diagiam, mtioduced in 1901 by Di R 
Mollici ', is of gicalci senicc in the solulion of practical pioblems 
By taking I'oi cooidmatcs the cntiopy and the total heat, Molhci 
consliucls a cluul which fiom tins point of MOW has advantages 
thai cnlillc it lo the fust place among dc\ ices for lepicscntmg 
giaphiCiilly I he Ihcimodynannc action of steam in steam-engines, 
01 of the woikiug fluids in iefi igeialing machines Its applica- 
lions m icfngciahon \\ill l)e dealt uilh in Ihc next ehaplti As 
icgaids steam il fuinishes Ihe mosl convenient \\a^ to mcasme 
Ihe heat-chop m tididbatic expansion, \vhatc\ci be the initial 
slale <is lo snpciheal, and conscqncntlv lo find Ihc gicalcsl 
lluoicheal oulpul thai is allamable when Ibc inihal picssmc and 
U mpc'ialmc, and Ihe final piessuic, aic assigned We 1m e seen 
I hat tins can be calculated when tables as complete as Callendai's 
aie available, and also that il can be found by the aid of an 
cnliopy-lcmpeialiiic chait on which lines of constant total heat 
have been diawn Bui the Mollier chait allows a giapluc solution 
lo be obtained with gicat du eel ness and case 

Foi practical purposes the Mollici 1$ chait is diawn so as to 
show only the steam boundaiv curve and the icgion immediately 
above and be-low it, but il is mshuclivc lo considei the complete 

* Jl Alollioi, "Ncuc Diagiamme zur lochnischon Waimclciluo," ZahrJnift (lea 
Veremea dcuhcher Inycmeure, 1904, p 271 See also IHR Nnie Tabellen und Dingrammz 
flir Wasaerdampf, Boriin (Jubus Spnngci), 1906 



122 



THERMODYNAMICS 



[en 



chart for watei and steam, which is sketched in skeleton foim and 
to a veiy small scale in fig 31 Theie ea is the watei boundary 
curve and bs is the steam boundaiy cuive The straight lines 
between them, such as ab and es, aie constant-pressuic lines, one 
of these (for P = 200 pounds per sq. inch) is continued across Lhc 
boundaiy into the region o bupeiheat, the ciuvc bb' icpieseiits 
the piocess of supeihcatmo at that pressure. The slope of smy line 
of constant piessure is a measiue of the tempeiatuie, for at con- 



800 




Fig 31 Molbci I(j> Cluil for WaLor and Sloam 



20 



btant pressuie dl = cZQ = 2V%and consequently rf//<7r/> = T In the 
wet legion the tcmpciatme along any line of constant picssme is 
constant, being the tempciatuic of satmation foi that piessurc, and 
therefore any constant-pi cssme line in that region is straight It 
ciosses the steam boundaiy without change of slope, but gradually 
bends upwaicls m the icgion of superheat as the temperature uses, 
foi its blope continues to be a measure of the tempeiatuie. 

The \vatei and steam boundaries aic connected, as in fig. 30, 
by a conjectural line bhiough the cutical point. The cnlieal point 



mj THEORY OF THE STEAM-ENGINE 125 

is not at the summit of this line, but at its point of mflectioif, 
which is also its point of maximum slope At the cutical point 
the continuous boundary curve, shown by Lhc broken line, would 
touch a cuive of constant pi ensure, and consequently its slope 
there, dl/d<^, is equal to the critical temperatuie, the absolute 
value of which is G3S The broken line is accordingly drawn to 
have a slope of 038 units of / for 1 unit of cf) at its steepest pait 
wheic, foi some distance, it is very ncaily stiaight*. 

Each constant-pi cssure line in the wet region may have its 
length between the two boundary curves divided into parts which 
express the dryncss q at successive stages in the pioccss of vapori- 
zation, ]ust as in the T(j> chait. Since the heat taken in up to any 
stage of that process is pioportional to g (Ait 70), equal distances 
along the line, coiiespondmg as the} do to equal mciements of 
total heat, coiicspond to equal changes of chyncss In this way lines 
of constant diyness aie dctcimmcd, sonic of which die shown in 
the sketch. 

It is useful to ha\ c a system oi lines of constant tcmpeiatnie 
diMMn in the icgion ol'snpeihcat two such lines aic shoun in fig 31. 
When they and I he constant-pie^siuc linos m thai legion have 
been chawn it is casv to maik Hit 1 point which corresponds to any 
assigned condition of the steam as to tcmpciatuie anel piessme. 
Thus b' is the 1 j)omt coiie'sponclmg to steam uilh a piessme of 
200 pounds, siipcihe^atcel lo 100 C 1 Then by dialing a vcitical 
sliaighl line thiongh the- point so found, \\e^ exhibit the pioccss of 
adiabatic expansion The lenglh ol Ihal line, elo\\n lo the final 
picssmc, nieasiue-s Ihc adiabalic heal-diop, and Ihcicfoic gncs a 
vciy simple and dncct means of finding Ihc gicalcst amonnl of 
\\oik ideally obtainable. horn a poiincl ol Ihe woikmg substance 
Thus I he 1 heal-diop in adiabalic expansion down lo a picssure of 
one pound per sqnaic inch is determined b> measuimg b'c on Ihe 
scale of / The position of c among Ihe- lines of constant di3iicss 

* Tlio Rlopc of tho boundaiy ciuvo, which is ( ) , w equal, at Uio critical point, 

\ " 7 ' I s 

to tlio slopo of the connUnl-prossuio lino which Luuc Iios li Uioio, nanu ly f J But 
) = T, mnco m any constant prossiuo clian^c dT = Td(/> Honr o at tlio critical 



point ( f 7 - ) -- T Houco also, at that pomt, ( 7 ) = ( ) But ( , ] .which 

1 U0/H \&p-Jt \^L \dl>J ri 

is the slope of the boundary curve m the enkupy-tompoiatmo chatt (fig 30), is 



zoio at the critical point Ilonce at the ontioal point , , J =0, tliat is to say 
the boundary curve in the /</> chart theio undoigoes inflection 



122 



THERMODYNAMICS 



cJnows how much of the steam is condensed by llus adKibaiie 
expansion. The advantage of a high vacuum, Lo which attention 
was diawn m AiL. 95, will he obvious fiom llic cflceb of the fin.il 
pressiue on the length of b'c. 

A thiotthng. f pioccs,s is represented by a hoiuonlal sliaii>hl line, 
suice / is constant Lines of consLant tcmpcialmc in I he supei- 
heated legion become neailv stiaight and liomontal aL very low 

O 1,1 

piessuies, for the behiu loin of the vapoui then appioxunates lo 
that of a peifect gas. 

800 ___________ 

~40QC 

. --- 



750 




=Q 7 

EntfOfjy (p 

15 16 \T \ 8 "l 9~ 
32 Mojhd's Cliari of Total JTcuil and E 



20 22 



A complete Rankmc cycle is shown by Ihc closed ligine cabb'cc, 
wheie ea is the heating oi the feed-watci, ab Ihefommdon ol slcain 
in the boiler, W its snbsequenL snpcrhcalmg, b'c iL^, adiabahc 
expansion to the piessuic of Lhe condcnsci, and cc ils, coiuhnsalion 
at that pipssure Foi the piaclical use of the diagram, ho\vcvei, 
theic ib no need to include the whole cycle Whal is wanted is 
the legion to the light, wneic the quality of the slc.uu IK fore 
and after expansion is exhibited, especially [he u-gion ['mm 



m] 



THEORY OF THE STEAM-ENGINE 



125 



/> - 1 5 to 2 and from I = 450 to 800, and by restricting the 
uhail Lo this i eg ion open scales may be used without making it 
unduly laige. 

Fig. 32 gives, in mmiatmc foim, a Mollier Chait for the useful 
region, showing, a few lines of constant piessure, also lines of 
constant Icmpeiature in the legion of superheat, and lines of 
constant dryness> in the wet legion*. 

102 Other Forms of Chart. Besides the foregoing diagiam 
Molliei mil educed anothci in which the cooidmates are the 



800 




100 
Fig 33 



Pi esiuni Pounds per ,y inch 
200 ' 300 400 "TOO 

Mollic'i's (.'luut of Total Heat and PUHSUIL 



picssme and the total heat A skeleton PI chait foi steam is 
shown in fig. 33 foi the icgion useful in piacticc. It has I he piopcity 
that lines of constant Lcmpeiatuie and lines of constant volume aie 
straight. It includes the steam boundaiy curve and a paitof the wet 
icgion below it, which is mapped out by lines of constant diyness 
as in the other chatts Tluough the wet region and the legion of 

* A chart of t,hi kind, exlubitmg Callenrlar's figuies on a scale large enough 
for practical use, has been diawn by Piof Dalby for his book on Stoarn Power 
(E Arnold) and may be purchased separately 



126 THERMODYNAMICS [en. 

superheat above, lines of constant volume aie diawn. They arc 
stiaight in the region of supeihcab, and sensibly stiaighL in the 
wet region, but they undergo an abiupt change of direction on 
crossing the boundaiy. (See Aits 208 and 209 ) 

Vaiious other chaits may be deviled by selecting for the two 
cooidmates othci pans of propeities fiom the list given in Ait 75. 
In any such chait the characteristics of the fluid arc exhibited by 
drawing systems of cim es, each of which icpiesenls the iclaLiou 
that holds between the two pioperties chosen foi cooidmatcs when 
the state alteis in such a manner that some thud property is kept 
constant By chawing seveial such systems of curves a compio 
hensive giaphical substitute for numeiical tables may be con- 
structed. The particular piopeities selected foi the coordinates, 
and for the cmves, may depend on the type of pi obi em 01 problems 
foi which the chait is wanted. Callcndar gives, as an adjunct to 
his steam tables, a chait m which the cooidmates arc the total 
heat and the loganthm of the piessmc. 

With lespect to all such devices it may be said that, so far as 
steam is concerned, the publication of full tables, which include 
the region of supeiheat, lender giaphic tabulation less necessaiy. 
It is now comparatively easy to find any icqmicd quantities 
chrectlv fiom the tables, 01 by mteipolation fiom them, with 
gieatei accuiacy than is leached in measuimg fiom a, chait But 
foi ceitam pm poses the giaphic piocess it, sufficicntlj'- exacl and 
more convenient All students should in any case make thcmseh cs 
acquainted with the entiopy-tempeia,Luie chait, and also with (he 
Molhei chait of entiopy and total heat the foimer because it will 
help them to undcistand cyclic pioccsics, the lallci as an insliu- 
ment for dealing with piactical pioblfins m steam cMgmccung and 
mechanical lefngeiation 

103. Effects of Throttling. We have ahcady .seen (Ail 72) 
that when a tluottlmg piocess is cained out nuclei conditions lh.it 
pi event heat fiom cnteung or leaving the substance the lolal 
heat / does not change. Lines of constant lolal heat on any of 
the diagiams accordingly show the changes in othci qiitiiililies 
which aie brought about by thiottlmg. It JS the piocess that 
occurs when a fluid passes through a "icducmg valve" 01 other 
constricted ounce such as the porous plug of the Joule -Thomt,on 
expeiiment (Ait 19) It is not what occurs when a jet is foi mod, 
as in the nozzle of steam tin bines In that piocess, which will bo 



rr] THEORY OF THE STEAM-ENGINE 127 

Icalt ^ith latei, the stieam of vapoin acqunes kinetic eneigy that 
nay be tinned to useful account, wheieas in thiottlmg, any 
inctic eneigy acquiied in passing through the constriction is 
mmcdiately dissipated by internal fuction. 

In a perfect gas throttling produces no change of temperatnie 
Ait 19), but in steam and other vapouis it produces a cooling 
ffcct which is measured as the fall of temperature pei unit fall 
>f pressuic under the condition that I is constant, or 

Cooling effect = ( - -) . 
\dPJx 

Values of this quantity for steam nuclei various conditions can 
>c deduced from Calendar's tables In steam that is highly 

o m/ 

upeiheated, especially at low picssuie, it is small, foi the con- 
lition of the steam then appioaches that of a peifect gas, but if 
he steam is saluialccl or only slightly superheated the cooling 
ffcct of thiottlmg is much gi eater Thus with steam at a pressme 
if 20 pounds per sqnaic inch, the cooling effect is only 0513 at 
00 C but is 338 at the tcmperatme of saturation These aic 
he falls of Icmpciatuic, due to thiottlmg, foi a chop in piessuie 
f one pound pci squaic inch The cooling clfect plays an un- 
(ort.mt pail in dclci mining the values of the total heal and othci 
aopcitics of ilic vapom, in the method used by CaHdidar 1 " 

the values given m his tables foi the total heat of 



* Callundar tabulates for steam a quantity (called by him >C) which is the 
>roduot of tho cooling eftcct and tlio specific heat at constant pies=uue It is a 
uniiUl}' of heat, namely the nuinboi ot ealonef) which would lu\e to be civ en 
D each Ib of tho thiottled steam to rostoio it, at constant pie-^me, to the tompeia 
uro it had before throttling, when tho amount of (hrottlmg is such that the piossuro 

f/7'\ 

tops hy one pound poi squaie inch Tt may bo written /> 01 K,, ( 7P ) it Js equal to 

\tU 1 1 

- ( ) and is independent of tho piossuro (as will bo shown later) The values of 

SO" or p which aio given in the (able for saturated steam theiofore apply alao 
o mipeihcntod steam at tho same temperature 
The cooling olToct C may bo found by dividing tho tabulated values of "SO" by 

ho specific hoat Tho specific hoat, which is ( - ) , changes only slowly with the 

omporature It may thcioforo bo found from the tables, for any given pressure 
nd terupoi atu ro, by noting tho difference between values of I at that pressure and 
t tompeiatnros above and bolow tho given tompoiature, and taking the amount by 
rhioh 7 olmngoE por degree Thus, for example, at a constant pressure of 20 Ibs 
ho rate at winch 7 changes with tho temperature is 509 calory por degree m 
lio neighbourhood of saturation For saturated &team of that pleasure "SC" is 
ivon as 172, hence tho cooling effect of throttling, por pound drop of pressure, is 
172/0 509 or 338, as has boon stated in the text 



128 THERMODYNAMICS [en. 

supeiheated slcam, it is easy to calculate how much the steam is 
cooled by any given drop of picssuie in throttling Let saluuited 
steam, t'oi example, at 200 pounds per squaie inch be throLLlcd 
down to a piet,suie of 20 pounds. The value of /, which icmains 
constant, is 609 T. At 20 pounds, the table shows that this value 
coiiesponds to a temperatuie of 163 8 The saturation tempcia- 
tme for 20 pounds is 108 9. The ongmnl tempera tin c was 19-A 3 
Thiotthng has theiefoie cooled the steam by 30 5, but at the same 
tune it has caused it to become supci heated to the extent of 54 9. 
The apparent paiaclox, that thiottlmg both cools a vapoui and 
supeiheats it, is due to the fact that when the pressure is reduced 
by throttling the saturation tempeiatiue has fallen more than the 
actual temperatuie has fallen. Hence satuiatcd steam is supci- 
heated by thiottlmg, and steam that is initially superheated 
becomes moie supei heated. Similaily, a mixtuie of vapour and 
liquid is paitially dned by throttling, it may be completely dncd 
and even supeiheatcd if there is not much initial wetness and if 
theie is a sufficient piessuie-drop This is illustiated m iig 30 by 
the line of constant total heat foi / = G50, which is cliawu paitly 
in the wet legion and paitly beyond it It bhows the cifccl of 
thiolthng on a wet mixtuie that contains (J S pci cent of water a! 
a press me of 500 pounds, the steam becomes chy when the pict,Mire 
is icduced to 37 pounds, coiiesponchng to the tcnipeiatuic of 128 
at which the line ciost.es the satuiation cui\ r coi steam boundaiy 

The piocess of throttling is still moie simply shown by hoir/ontal 
lines (/ = constant) in the Molher diagicim (fig 31) By thawing 
such lines tlnough the points on the satuiation ciuvc for 2' 1 
and P = 15 it will be t>een that 12 per cent ol watei can bo 10- 
moved fiom steam at 200 pounds pressmc by throttling il down 
to 1 pound, 01 fully G per cent, by thiottlmg it to atmospheiic 
piessme. Smnlaily, it is easy to liacc the extent lo winch liquid 
will cvapoiaie in escaping tlnough a thro I tic-valve fioin a i eg ion 
of high pressure to a icgion of lowci picssuic. 

The method of chymg by thiottlmg has been applied as a means 
of determining the peicentage of watci picscnt in steam For I Ins 
puipose a device is ui,ed that is called, lather nuppropnalrly, a 
"thiottlmg calorimeter." Its essential feature is a pipe through 
which a sample of the steam to be tested can be passed, containing 
within it a diaphiagm with a pin-hole oiifiec, 01 a tluottle-valvc 
01 poious plug, tlnough which the steam has to pass. There are 
presbine-gauges on both sides, and a theimometei to read the 



nij THEORY OF THE STEAM-ENGINE 129 

temperatuie of the steam immediately after passing the obstruc- 
tion. Both parts of the pipe must be thermally insulated, so that 
no heat is lost, nor conveyed by conduction fiom one part to the 
othei The amount of steam passing, which may be regulated by 
means of another valve beyond the obstiuction, should be such 
that the steam aflei tlnottlmg is appieciably supeiheated, in oider 
that no wetness may be lefl in it, complete dnness is ensuied by 
seeing that the tempciatuie aftei throttling is someAUiat highei 
than the satmation tempciatuie Then fiom the tables we find I' 
the lotal heat which concsponds to the tempeiatme and piessuie 
as observed after tlnottlmg Since thcie has been no change m 
the total heat, this must be equal to I w + gL, wheie these quan- 
tities icfei to the state befoie tlnottlmg Hence the initial chyness 
is found, namely r> r T' r 

' J _ *_ _~ l w _ - 1 ~ J-w 

Q ~ T. ~ T T ' 

Ms J. ^ J. w 

In practical applications of this method a poious ping is to be 
pu'lcncd to a thiotllc-'v al\e because Lhe thcimomctei can be 
placed close to it and the lempciattue mcasuied aftci the 
thioltlecl sticam has lost its kinetic uicigs and bcloie it has 
stilfcicd loss ol heat It is dilncull in am case to scenic that the 
sample tested by any such appaialns shall be piopulv icpiesenta- 
h\c, in icspcct of the nioistine it caincs, and consequent^ little 
leliancc c<m be placed on tests thai me earned out b\ duelling 
,i poil ion of a steam supply into a thioltlmi> caloi nuclei, as 
a means of deleimming the gcncial \\clncss of the supply. 

104 The Heat-Account m a Real Process The pioccsses 
which luiAc been consideicd in lhischai)lei as going on in a steam- 
engine aic ideal in the sense that they have been assumed to be 
a (hail ic) Dial that is to say, thcie is no tiansnnssiou of heat to or 
fiom the woikmg substance except what is oiigmally taken fiom 
the source 01 finally iqccted to the icceivei , m all the mteimeehate 
opt rations the working substance has been enclosed in vessels that 
aie assumed to transmit no heat. The assumed piocesses arc also 
ideal in the sense that they are internally reveisible The piocess of 
throttling, which is a typically mcversiblc piocess, did not occur 
in the ideal engine cycles In dealing with it also, howevci, we 
postulated adiathermal conditions, it was assumed in the argument 
of Ait 72 that no heat passed by conduction thiough the con- 
taining walls to or fiorn the space outside. 

9 



B T 



180 THERMODYNAMICS [<". 

Discarding these limitations we may now diaw up, 111 general 
teims, a balance-bheet or heat-account foi any real piocess, winch 
will include thermal loss to the space outside and also meuTsible 
actions within the engine 01 other apparatus 

Whethei the apparatus considered be an engine cyluulc'i, 01 I he- 
series of cylinders of a compound engine, 01 a tuibinc, or a I In oil - 
ling device, we may in all cases compaic UK- stale of Ihc Hind Jil 
entiy and at exit, as foi example in the admission pipe- of an engine 
and in the exhaust pipe We imagine a steady (low ol' llu- woi king 
fluid thiough the apparatus At entiy let ils pubsuie be* P { , ils 
volume (pei Ib.) F x , and its mteinal eneigy 7i' L At exit Id. ils 
pressure be P 2 , its volume F 3 and its internal encigy A' 2 . To nuiUc 
the companson complete we may wnte K I'oi Ihc kinetic energy 
(also per Ib.) of the stieam as it enteis, and A' 2 foi ils knulic 
energy as it leases In passing thiough the apptuatus flu- Jluid 
will, in geneial, do exteinal woik, and also lose by conduction 
some heat to exteinal space. Let W rcpicscnt, in thc-imal uiuls, 
this output of woik, and let Q L rcpiesent the heat lost by conduc- 
tion to exteinal space, both of these qiuin Lilies (like Ihc olluis) 
being leckoned pei pound of the Jluid that passes thiough 

Each pound that enteis the appaiatns rcpiesenls a sii|)ply of 
energy equal to K t + E l + AP^V^ , 01 E l is the internal cncigy il 
cames, and PJ /r l is the woik done by the fluid behind in pushing 
it in. But E-L + AP 1 V 1 is equal to I l} the total heal pc-i pound ol 
the fluid m its actual state at cntiy Sinulaily, each pound lhal 
leaves theappaiatus icpiesents a i ejection of cnejgy amounting lo 
K z + E, + APjFz, foi E 2 is the mteinal cneigy which (In Hind 
cames out, and P,F 2 is the woik spent upon il by lh< Iliud lu-lniid 
in pushing it out E, + AP 2 V Z is cqu.il lo /_>, (he lol.il he;il p< i 
pound of the fluid m its actual stale at exit IK nee, by Ihc con- 
seivation of eneigy, foi the apparatus as a whole, 

/L! + /i = /iT 2 + / 2 -I- W + Qi 

The terms on the left of tins equation icpiescnl the eneigy (h;U 
enters the apparatus, the tcims on the right show how il is disposed 
of in the issuing stieam, in output of uselul woik, and in lenluige 
of heat 

The teims K t and K z are usually veiy small, except when (he 
apparatus is one foi foimmg a steam jet, in which case A'., is (he 
useful teim this will be considered m a lalcr chapter When the 
change of kinetic eneigy m the stream is piactically negligible, 



uij THEORY OF THE STEAM-ENGINE 131 

as it is between the admission pipe and exhaust pipe of an engine, 
wehave Ii-Ii+W+Qt. 

And when, in addition, the appaiatus does not allow any appieci- 
ablc amount of heat to escape to the outside (Q t = 0), we have 



Thus means that when Iheie is a stead}' flow ot a Avoiking sub- 
stance through an> theimodynanuc appaiatus, the output of woik 
is measmcd by the actual Hcat-Diop, ivhelhei the internal action 
is 0) 'is not levasible* piovided theie is no loss oi heal to the outside 
by conduction thiough the walls 

The actual heat-chop must not be confused with the achabatic 
heat-chop, which is Lhe diffeiencc between l and that value 
which the total heat would reach if theie were achabatic expansion 
to the exit prcssuie P z The actual hcat-diop I : /, is identical 
with the achabatic hcal-diop only when there is no loss of heat to 
the outside and when, in addition, the internal action is wholly 
leveisible 

Any uicveisiblc fcatinc in the mtcinal action will mciease /, 
abo\ e the value which \\onld be i cached b\ achabatic expansion, 
iind will consequently diminish Lhe output of woik 

In the cxlicme case of a lluotthng piocess Ihcic is no output 
ol woik, and theicfoic J> = f 1 , pio\ ided theie is no lo-,s ol heat to 
the outside Am loss ol heat lo the outside in a thiollhng piocess 
will make /., coiicspondmgK less foi we then lia\ c /, = / x Q, 

The losses ol theimod\ nanue cllecl in a util engine which make 
ll'lcss lhan the ideal output, namch the \ aluc coiKxpondmg to the 
.uliabalic he.il-diop, ansc paith horn loss of heat lo the outside 
and pailh horn I wo kinds of uieveisible mteinal action One of 
Ihese two kinds is mechanical, the othei is thcimal In the 
mechanical kind, I he acLion mvoh cs (hud hiction within the 
woilung subslanee It is ol Ihe same natuie as that which occuis 
in Ihrotlhng llieic is uie\ cisiblc passage of the uoiking substance 
fiom one pait ol the engine to anothci whcie the picssuic is lo^ei, 
as foi instance the passage of si cam thiough somewhat eonstiictcd 
openings into the cyhndci, 01 its passage, on icleasc aftci incom- 
plete expansion, into the exhaust pipe, with a sudden diop of pres- 
smc or again, there is the same kind of ineveisibility in a tin bine 
in the fnctional losses that attend the formation of steam jets or 
in the friction of the jets on the turbine blades These are all 
instances of mechanical meversibility. In the second kind of 

92 



132 THERMODYNAMICS [c-ii. ni 

ureversible acLion there is exchange of heal bclwccu (he working 
substance and the internal suilace of Lhc engine walls. The hoi 
steam, on admission to a cyhndei which has, jusL been vacated by 
a less hot mixtiue of steam and water, finds llu> surfaces coldci 
than itself. A pait of it is accordingly condensed on I hem, whu'li ie- 
evapoiates after the prcssuiu has fallen through expansion. This 
alternate condensation and ic-evaporation involves ,i consult rable 
deposit and iccoxci}' of heat in a manner Ihal is not re \cisible, 
foi it takes place b} contact between ilmd and mcl.d al dilUicnl 
tempeiatuies. The action may oeciu withoul loss of lii'al It) the 
outside it would occui, for instance, in an engine wil h a, ceinducl mg 
cyhndei covered extcmalty with a "lagging" of non-conducting 
matenal. Its effect, like that ot thioltling 01 lluid Jnclion genci- 
ally, is to leduce the output of work below Ihe limit Ihal is 
attainable only in a le^-eisible piocess, and it does this by making 
the actual heat-chop I I / 2 ^ css Ihan the achabalic heal-diop 

The equation W = I I z takes account of both kinds of uie- 
veisibihty of the effect of theimal exchanges wi I Inn I he appai.ilus, 
as well as of any throttling or fuchonal effects in I he aetion of I lie 
woikmg substance But it docs not take aeconnl of heal losl lo 
the outside, and foi that the teim Qz has to be deducted, m, living 

W = ll ~ /2 - l 

The full statement of the heat-account in a teal piocess may be 
expiessed as follows When thcic is a steady (low of a woikmg 
substance thiongh any thermodynamie appaialiis, the out put ol 
woik ib measuied by the actual heat-chop fiom c-nlMiiee lo (\il, 
less any heat that escapes by conduction lo Ihe outside, and h ss 
any gain of kinetic eneigy of the issuing sticam ovei (he c nUim<r 
stieam, 01, in symbols 



all these quantities being expiessed in Iheimal nnils, and reckoned 
per unit quantity of the woikmg substance. 

This equation also applies to reversed heat-engines, or he.il- 
pumps, which will be consideied in the next chapter, but in Hum 
the quantity W is negative- work is expended on the machine 
instead of being produced by the machine. In such machines Q, 
is also geneially negative, for as a rule the apparatus is colder Ihan 
its surroundings and the leakage of heat is imvards 



CHAPTER IV 

THEORY OF REFRIGERATION 

105. The Refrigeration Process. Refrigeration is the le- 
moval of heal from a body that is colder than its surioundmgs 
In cold stoiage, foi example, the contents of a chamber aie kept 
at a tcmperdtuie lower than that of the an outside, by extiactmg 
the heat which continuously leaks in thiough the imperfectly in- 
sulating walls, To maintain a icfi iterating pioccss lequnes ex- 
pendituie of eneigy It is generally done by means of a mechani- 
cally dnven heat-pump, working on what is essentially a icversed 
heat-engine cycle. It may also be done by the direct use of lngh- 
temperatuie heat without mtcimcdiatc conversion of that heat 
into woik We shall consiclci Jatci the duect application of heat 
to effect icfngeiation, but shall in the fust instance tieat of re- 
tiigciatmg machines dm en by the expendituie ol mechanical 
powei. 

Any pioeess of refngci Alton imohes the use of a woiking 
substance which can he made to take in heat at a low tempeiatuie 
and dischaigc heal at a highct temperature Tlic heat is dischaiged 
by bcjng <>i\ en up to the an outside 01 to any walei that is available 
to icceive il The piocrss is a pumpmg-up ol heal tioin Ihc level ol 
teinpcialme ol the 1 cold bod\, at \\lnch it must be taken in, to the 
lexcl ,1,1 \\hich it max be dischaiged These levels should be as ncai 
toyclbei as is piacticablc, in oidct that no unucccssaiy woik may 
be- done in ol her uouls the action of the woiking substance should 
be confined to the nauowcst possible lange of tcmpcratuic The 
U'lnpeiatnic of dischaigc should be no highci than is necessaiy to 
ncl lid of the heat, and the kwu limit should be no lowei than 
wtJI ensure Iransfei of heal into the icfiigciahng substance fiom 
the cold boety whose heat is to be extracted 

Let r l\ be the tcmpciatiuc at which heat is dischaiged and T 2 
the tempcuiluic at which it is taken in fiom the colel boely Con- 
sielei a complete cycle in the action of the woiking substance. Let 
QJL be the quantity of heat which is discharged and Q 2 the quantity 
which is taken in fiom the cold body; and let W be the thermal 



THERMODYNAMICS [en. 

equivalent of the woik spent in diivmg the lefngcialing machine. 
Then, by the conservation of eneigy, 

The useful lefngeiatuio effect is mcasmcd bv Q ? , and the Ck co- 
efficient of peiiormance," which is the latio of that effect lo the 

Q 

woik spent in accomplishing it (Art 4) is , ~ . 

106 Reversible Refrigerating Machine. We have first lo cu- 
qune what is the highest possible coefficient oJ peifoimancc when 
the limits of tempeiatme 1\ and T 2 aic assigned We know by 
the pimciple of Carnot (Aits 33, 39) thai when heat passes down 
from Tj to T , thiough a heat-engine, the ideally greatest clliciuicy 
in the conversion of heat into woik is obtained when I he engine is 
thermodynamically reveisible In that case 



The output of woik W is Q 1 Q 2 . Hence the ideally giealost 
output of work is i elated to Q 2 , the heat i ejected a I the lowci limit 
of tempeiatme, by the equation 

rvQA^-TJ 

T 2 ' 

A coiiespondmg pioposition m the thcoiy of icfugeiahon is lh.il 
the ideally gieatest coefficient of peiioimance of a icfiigcinhiig 
machine, woiking to pump up heat fiom T lo 2\, is obtained when 
the machine is theimodynamically ^eisiblc. In lh.il disc the 
same lelation holds, namel} 



and the amount of woik W Mhich is spent in duving Hie machine 
(and is> equal to Q x - Q,) is iclated to Q.> by the equation 

w _^( r l\-T 2 ] 
~ T 

* 2 

In othei woids, the gieatest amount of woik that is theoretically 
obtainable in letting heat pass clown through a qivon range df 
tempeiatme is the least amount of woik thai will suffice lo pump 
up the same quantity of heat thiough the same range 

To show that no lefngeiatmg machine can be more cfucienl than 
one that is reveisible, we shall use an aigiimenl like thai of Ait W 
Let E, fig 34, be a leveisible lefngeiatmg machine, leverscd and 



IV 



THEORY OF REFRIGERATION 



135 





Q, 



= Q 5 



R 





theiefoie serving as a heat-cngnie. It takes a quantity of heat, 
say Qi> fiom the hot body 
and delivers a quantity Qa 
to the cold body, converting 
the difference into work. 
Let all the work W which 
it develops be employed 
to drive a lefngeiatmg 
machine R, and assume 
that there is no loss of power 
in the connecting mech- 
anism Accordingly the two 
machines, thus coupled, 
fonn a self-acting combina- 
tion. 

If it weie conceivable 
that the machine R could 
have a gieatcr coefficient 
of pei loi mance than the 
ie\eisible machine E, that il0 

would mean that the latio of Q 2 to II' would be gieatei in R than 
in E Hence (II' being the same lor both) It Mould take moie heat 
fiom the cold body than E gi\ es to it, and R Mould also gi\ e moic 
heat to the hot bodj than E takes fiom it The icsiilL Mould be 
a continuous tiansfci oi heat fiom the cold bod\ to the hot body by 
means ol j. puicly sdf-aclmg agency This would be conliaij to the 
Second Law ol Thumod\ nainics we conclude theieloie that no ic- 
t'rigeiiitmg maclunecan havea highei coclhcientol peifoimanccthan 
aicveisiblc machine woikmg between the sainclimilsoltempeiaturc. 

It follows that all ic\cibible icfiigeiating machines, MOikmg 
bclwcen tlie same limits of tcmpeiatuic, ha\c the same coefficient 
of peifoimance. It also follows that the value of this coefficient 
is thai which would be found in a icvcised Cainot cycle, namelj 

W = r J\ - T, ' 

This is the ideally highest coefficient it mcasuies the peifoimance 
of what may be called a pci feet icfiigci al ing machine The coefficient 
of peifoimance in any real machine is neccssaiily less, foi the cycle 
of a real machine fallt, short of icveisibihty 

107 Conservation of Entropy in a Perfect Refrigerating Pro- 
cess We saw in Art. 45 that a pcifcct, or reveisiblc, heat-engine, 



136 THERMODYNAMICS |rn. 

such as Carnot's, may be legaidcd as n, device which Inmsl'ci'- 
entropy from a hot body to a cold body wiLlioul, allerin^ UK 
amount of the entropy so transfcucd, allhou^h the ainounl of 
heat which enters the engine it, grcatci than the mnotiHl of Ju-nl 
which leaves the engine. The entropy to ken liom Uu; hoi body, 
namely QJTj^, is equal to the entiopy given lo I lie oold body, 
namely Q. 2 JT Z \ it may be said to pass Ihi'ough Iho engine wilhoiil 
change, though the heat that passes through is nduetd in I he 
piocess by the amount which is con veiled mlo woik, lunncly, by 
the amount Q L Q . 

Similaily a peifect, or reversible, rcfngei'alinn niachmo or lx.il- 
pump may be regaidecl as a device which hansleis enliopv hoiu 
a cold body to a hot body without alluing Uie amount <>| Uu 
entiopy so tiansfeued, although the amount oJ lu.il whu-li eiilt is 
the machine is less than the amount which leaves Ilio machine. 
The action is in eveiy particular a, icversal of lh.il of the ju idcl 
heat-engine. Entropy to the amount QJ f J\ is I. ikui fioin Ilu cold 
body, and entiopy to the equal amounl <y,/7', is ^i\cii lo I he 
waimei body to which heat is disch.nned The .iinoiinl of lu.il 
wbch is pumped up mcieascs ftom Q, to Q { in I IK piocc-ss, because 
an amount of woik equivalent to Q L Q, is cvpcndcd in diainy 
the machine and is conveited into heal williin I In niachiiK 

108 Ideal Coefficients of Performance The follou HIM lalih 
shows the \alues of the cocflicicnL of pcifoiiuanec in tl pulid 01 
leversibleiefrigeiatmopioccss, loi vanoiis lanyeM)! (cinpci.ihin 
These aie ideal Homes, leprcscnhno <L lli(. ,clu'.il liniil u|,i<|, 
cannot be i cached in piaclicc Thouoh I hey ulalc lo coiidilions 
of levcisibihty which aic nol fully allaiuabk- in a ical niacluiK , 
they illustiatecleaily lhcpiacliCiilini]K)i lance <l in,ikiii K Ilu i.mn/ 
of tempeiatuie as small as possible, by lakini> in UK IK at til a l< m- 
peiatuie no lower than can be helped and by disdi,irni ni , il nll< i 
the least piacticable use 

Coefficients of Pafoimancc of a Pojnl Ktjn^itilnm Mnclnn, 



00 
I 

I I 

I 

n n 

7 J 



Lo\\u Ijinifc 
of teinpeitituie 




U|>])u limit, ol |,( 


l!ll|)( 1 l( 
[|,,\ 


(Centigiade) 


10 


20 J 30'" " 


(in; 
40" 


- 20 U 


8 I 


<> .{ fi 1 


1 2 


- 15 


108 


7i r>i 


1 7 


- 10 


Li I 


So , . 
3 t) 


r> '{ 


- 5 


179 


107 77 


It 





273 


13 !) I 


(! N 


5 


55 b 


18 fi [] l 


7 (1 ) 


10 





28 3 14 1 


!) 1 



ivl THEORY OF REFRIGERATION 137 

The importance of a nairow range of temperature in rcfrigciation 

is fulther illustrated by T 

F~ f 1^ 

fig. 35 It gives the en- ' ''< _ , , 

00 A Lf.' WV,</*,M < i &/,*,/,, ' ' \ (m 

tropy-tempeiature dia- gj ^ ^ 

grams of tliiee reversible -3 B **^ ^ -^ T z (b) 

i efngeratmg processes, m 03 \ 

all of which the upper g" Q ^/^.^ T 2 (c) 

limit of tempciatuie (2\) ^ 

is the same, and the 

same amount of v^ork is 

spent. Each of the three Entiopy D 

supposed piocesses is F J g ^ 

ideally efficient il is a icveiscd CarnoL cycle, and its cntiopy- 
tempeiatuie diaijiam is a icctangle. The aiea ol the lectangle 
repicsenls the woik spent, and the aica under it, down to the 
absolute zcio of tempciatuie, icpicsents the amount of heat that 
ib Liken fiom the cold body, and llieitl'oic mcasuies the leingeiat- 
mg clfccl The tlucc pioccsses foi \\lnch the diagiam is sketched 
dillei only in the temperatmc T 2 oi the cold body fiom which heal 
is cxhacted Thai Lcmpciahuc is iclativch high in the inst case 
(a), lowci in case (b) and ]oA\ei still in case (t) The iciiigciahng 
cflccl is measuied by the aiea^li) in the fust case, by CD in the 
second, and In CD in the thud The lesull of lo\\eiino T 2 is ACI\ 
appaient, in leducing the amount of icfiigciation thai is ulcalh 
capal>lc of being done In a gi\ en cxpcndilnic of \\oik 

109 The Working Fluid m a Refrigerating Process The 
\\oikmg subslancc in ,i ulii^eiating c\clc may be a yas \\hich 
icmains gaseous tlnoiiglioiil, such as an. Moie commonly it is a 
(hud which is ,iltcinalcl\ coiiflcnsed and c\apoialcd Dining 
cxapoialum <il a Jow picssuic the lluid takes in heal fiom the cold 
bod} it is then compicsscd and ones out heal m becoming con- 
densed at a iclativcly high picssuic The selcclmn of the fluid is 
govcined by piaelical consideiations AValci is used m some cases 
but a sciious diauback to its use is the vciy laigc \olume and low 
pressure of the vapoui at low lempeiaLmes. Thcie aie obvious 
advanlages m using a fluid whose vapoui -pressuie is neithci mcon- 
vcmcnlly small at the lo\\ ei limit of tempciature noi mcoin cmcntly 
large at the upper him I The fluids most commonly used are 
ammonia and cai borne acid. Ammonia has a veiy convenient 
lange of vapour-pressure throughout the lange of temperature with 
winch we are concerned in piactical refngeration, With caibomc 



138 THERMODYNAMICS [cir. 

acid the vapour-prcssuie is considerably higher, the culical point 
is leached at a bempeiature that may come within I lie ninge of 
opeiation, and the theimodynamic cfllcicncy is somewhat less. 
Notwithstanding these objections caibomc acid is Jicqm-iilly pre- 
feired, especially on board ship, where it is moic hainik'ss slioultl 
any of the fluid escape by leakage into Ihc room ecu laming I ho 
machine. Foi use on land, especially whcie I ho highest Iheimo- 
dynamic efficiency is aimed at, ammonia is usually chosen Olhei 
fluids with lowei vapom -presumes arc occasionally used, such as 
sulphurous acid, ethyl chlonde, and methyl chloride. 

no. The Actual Cycle of a Vapour-Compression Refri- 
gerating Machine. If the icversed Cainot cycle were acliuilly 
followed, the choice of woikmg fluid would make no dilfeienco lo 
the efficiency the coefficient of perfoimance for any lliud would 
have the value shcmn in Ait 106, namely T,/('J\ - T,} Bui a 
part of the icveised Carnot cycle is omitted in practice, wilh (he 
lesult that the coefficient is i educed, and thecxlenl ol Ihe mlucliou 
depends on the natuie of the fluid, it is grcalci 111 caihomc acid 
than in ammonia 

To cariy out a reveised Carnot cycle, wilh srpauilc- organs foi 
the successive events which make up the cycle, would icqum- 

(1) A compiession cvhndei in which the vapoiu is compu ssc d 
fiomthepiessuiecoirespondmg to T, to IhcpicssmeootrcsMondiix. 
to T! " ^ 

(2) A condenser in which it is condensed a I 'J\ A typical form 
of this organ would be a sinfacc condcusci in ulnch [lie \\mkiM 
fluid gives up its heat to circulating watci 

(3) An expansion cylmdci in wluch it expands fiom V, lo 7',. 

(4) An evaporatoi in which il takes up heal ,il T, IK.III tlu- 
cold body fiom which heat is to be cxtiacled This x c ssr! Is some - 
times called the "refngeiatoi." 

In neaily all refiigeiatmg machines the expansion cyl.ndn is 
omitted foi icasons of piactical convenience, and I lie lluid sln-ains 
fiom (2) to (4) through a thiottle-valve with an ad,uslal>l( opening, 
called the ' legulaLoi" 01 "expansion-valve" In pass,,,., ll.o c-x' 
pansion-valve the pies.uie of the woikmg fluid falls lo IhM of Ihc 
evapoiatoi its tempeiatuie falls lo T, and pad of ,l | immit . s 
evapoiated beloi'e it begins to take in heat fiom UK- c-old body 

The omission of an expansion cyhndc,, w.lh the snbslilulion lo, 
it of an expansion-valve, lc duces the cocflicicnl of peifonaanoc for 



IV] 



THEORY OF REFRIGERATION 



139 



two reasons The woik which would be reco% eiecl in the expansion 
cylinder is, lost, and also the lefngerating effect in the evapoiator 
is reduced, for nioie of the liquid is vaporized in the act of sti earning 
through the expansion- valve than would be vaporized in achabatic 
expansion, consequently less is left to be evapoiated by subse- 
quently taking m heat fiom the cold body. The loss of efficiency 
from these U\o causes is not, however, very impoitant undei 
ordinal y conditions To omit the expansion cylinder is a consider- 
able simplification of the machine, all the moie as the effective 
volume of such a cylinder Avould need adjustment i datively to that 
of the compression cylmdei in ordci to sccuie the best effect undei 
vaiymg conditions as to the limits of tempeiatme Rather than 





Fitf JO Ui<riin- ot a V ijiom ( otiipi< s-mn Ki li in( ml m<j, M.uluiH 

inLioducc llus complication it is woilh while io make a slight 
sacuhce ol Ihcimodynannu ciricKiicy 

In the usual type of vapoiu-o'oinpixssion icli isolating machine, 
aecoidmglv, the expansion cylinder is onulled, and Hit oigans are 
those shown diagrammalically in fig 3G They aie, (1) Die com- 
pression cylinder /?, (2) a condensci A such as a coil of pipe, cooled 
by ciiculalmg walcr, in which I he working substance is condensed 
undei a iclalively high ])iessiue and al the upj)c i i linul of tcm- 
pciatuie T I} (J3) an cvpansion-valvc 01 icgnlaLoi 7? Unough which 
it sti earns fiom A to C, ( 1) the evapoiatoi C\ in which it is vapoiiml 
at a low picssuie by taking in heat fiom the cold body at the lowci 
limit of temperature The e\ aporatoi may foi instance be a coil 
oi pipe taking in heat from the bin rounding atmosphcic of a cold 
chamber; often it is a coil suiiounded by cold circulating brine 



140 



THERMODYNAMICS \ni. 

which serves as a vehicle for conveying heat Lo the woiking sub- 
stance liom a cold chambei 01 fiom cans for ice-making 01 olhoi 
objects that aie to be lefngerated 

The action of the compiession cjdmder is shown by the indicator 
diagram, fig 37, m the same figuie. During the ibrvvaid stroke of 
thecompiessoi the valve leading to A is shut and that leading from 
C is open. A volume V^ of the woiking vapoiu is taken in fiom C 
at a umfoim piessme coiiesponding to the lower limit 2' 2 . In 
most actual cases what is taken in is not diy-sal mated vapour but 
a wet mixtuie, the wetness of which is regulated by adjusting the 
expansion valve R. This is in ordei that the subsequcii t conipi cssioii 
may not produce much (if any) supeiheatmg It is possible lo 
make the compression wholly "wet" by taking in a sufficiently wel 
mixtuie moie geneially the expansion-valve is adjusted so th.it 
the vapom is modeiately wet to begin with, and becomes s 



\/ 



JFig 37 Indicatoi Diagiam of Compiession Cylmclpi 

supeiheated by compiession. At the end of the I'oiwaid sliokc UK 
\alve leading fiom C closes and the piston is foiced to mo\c b^ck 
compressing the vapom 01 wet mixtuie in Ihe e\linclei until Ms 
piessuie becomes equal to that in A Tins compicssion i educes 
the volume of the fluid in the cylinder to V^ The valve lejidmg lo 
A then opens, and the back-stioke is complclcd nuclei* a nniloiin 
piessme uhile the woiking substance is dischaiged mlo A <iml 
condensed theie The valves of the compicssor aic spimg v.ihis 
wliich open and close automatically in consequence of Llie ch;ingc-s 
in piessme, and are situated in the covci of the c^lmdci in suc-li 
a mannei as to make the clearance negligibly sm.ill To com])U It- 
the cycle, the same quantity of woiking substance is allowed lo 
pass dnectly from A to C thiough the expansion-valve A'. This 
step is not leveisible (Ait. 22) 

The tempeiature T l at which condensation takes place, is in 
piactice necessarily a good deal higher than that of the eiiculalmn 
watei by which the condense! is kept cool, for a laige arnouul of 



IV] 



THEORY OF REFRIGERATION 



141 



heat has to be dischaiged from the condensing vapour in a limited 
time. But it is impoitant that the condensed liquid should be no 
warmer than is unavoidable bel'oie it passes the expansion-valve. 
Accordingly the condenser is ai ranged (sometimes by the addition 
of a separate vessel called a "cooler") so that the condensed liquid 
is brought as neaily as possible to the lowest tempera tme of the 
available watei -supply before it passes the valve, though it may 
have been condensed at a considerably higher tempeiature. The 

b 



d 



g f (( h 

Pig 38 Tho Vdpoui Completion Cyclo, u-inn Ammonia 

I 




Fiy JO Tbo Vapoiii Compicssion Cyolo, using Carbonic Acid 

advantage of I his will be obvious when we considei, in the next 
u tide, the theimal cflects of each step in the cycle 

in. Entropy-Temperature Diagram for the Vapour-Com- 
pression Cycle. The complete cycle is exhibited m the entiopy- 
bemperature diagiam of fig. 38, which is diawn for ammonia as 
\voiking substance, and fig 39, which is drawn foi carbonic acid 
Tlieie dg and ch are portions of the boundary cuives. The point a 
represents the condition of the mixture which is drawn into the 
2ompression cylinder, when compression is about to begin; its 



142 THERMODYNAMICS [ CI1 - 

wetness is measured by the ratio ah/gh. The line db lepiesents 
adiabatic compression to the pressure of the condenser. The next 
piocess consrsts of cooling and condensation at this constant pres- 
sure it is made up of thiee stages, be, cd and de In the fiist stage, 
be, the superheated vapom is cooled to the tempeiature at which 
condensation begins, in the next stage, cd, the vapour is completely 
condensed, in the third stage, de, the condensed liquid is cooled to 
the lowest available tempeiatuie before it passes the expansion- 
valve. The lines be, cd, and de form paits of one line of constant 
pressure In fig 38 de is practical^ indistinguishable fiom the 
boundary line, but in fig. 39 the 
distinction is veiy appaient be- 
cause we aie there dealing with 
a liquid that is highly com- ^ 
pressible in consequence of its el \ 

nearness to the critical state. A \ 

The line efrepiesents the pro- ' \ 

cess of passing thiough the f 
expansion valve, m which the 
piessuie falls to that of the 
evaporator. This is a throttling 
piocess, for ^hich/ is constant 
(Ait. 72) ef is theiefoie a line 
of constant total heat, its 
direction changes m fig. 39 m 
crossing the boundary cui\e. 
Bypassing the expansion- vah e 
the wo] king substance comes 
into the condition shoun by 
the point / The proportion 
which is converted mlo vapour 
by the meie act of passing 
the valve is sho\Mi b}' the ratro 
gf/gh. Lastly we have the 
process of cflcctn c evaporation 
when the substance is uscfidly 
extracting heat from the brine 
or other cold body by evapo- 
rating m the refrigerator This p 40 n 
is represented by the line /, 
during which the d^ness changes from gfjgh to ga/gh 



iv] THEORY OF REFRIGERATION 143 

The refrigerating effect, that is to say, the amount of heat taken 
in from the cold body, is represented by the area under the line 
fa, measured down to a base-line corresponding to the absolute 
zero of temperature, namely the area mfan (fig. 40) 

The amount of heat i ejected during cooling and condensation 
of the vapoui and subsequent cooling of the condensed liquid, is 
the aiea under the lines be, cd and de, namely the aiea nbccleo 

The thermal equivalent of the woik spent in canymg the woikmg 
substance thiough the complete cycle which is simply the work 
spent on ib in the compiessoi is the diffeience between those two 
quantities, namely the aiea nbcdeo minus the area mfan. It should 
be noted that the work spent is not measuied by the aiea abcdefa, 




Fi# 41 < yU<> fin (Jaihoiuc Acid, \\ilh compiLision above the (Jnfica.1 Pressiue 

enclosed bv I he lines which icpicscnt the complete c\cle, because the 
cycle includes an mcveisible slop ef (sec Ait 51) In consequence 
ol that the woik spent is giealei than the enclosed aiea by the 
amount ocfni 

As a fuilhei example we may take a compression piocess (fig. -41), 
with cai bomc acid foi woikmg substance, m which the tempeiature 
of the cooling water is so high that the piessure dining cooling is 
above the critical piessuie The line be is accoidmgly a continuous 
curve lying entirely outside of the boundary curve. The woikmg 
substance passes from the state of a supeiheated vapour at b to 
the state at e without any stage coirespondmg to cd in fig 39, m 
which it is a mixture of liquid and vapour As befoie, the refn- 
g-eiatmg effect is measured by the area under fa. the heat rejected 
to the cooling water is measured by the aiea under be the difference 



THERMODYNAMICS [en. 

between these two quantities measuies the woilc spent, and is 
greater than the area of the closed figui e ale/a by the area undci 
the me \eisible step ef. 

112. Refrigerating Effect and Work of Compression ex- 
pressed in Terms of the Total Heat While it is ins Line Live 
to state, as in the pieceding article, the lefiigcratmg eflccl, the 
work of compiession, and the heat lejecLccl, in teims of areas on 
the entropy-lempeiatuie diagiam, it ib much moie useful, i'oi 
purposes ot piactical calculation, to expiess these as follows m 
teims of the total heat of the substance at the vanous slaves of 
the opeiation. 

The lefngeratmg effect, that is to say the amount of heal laken 
in fiom the cold bod)'-, is / - I f , wheie I a is the total hciL a. I a 
and Ij is the total heat at/. This is because the (icveisiblc) opera- 
tion fa is effected at constant piessuie (Ait 68) Foi Ihc s.ime 
leason the amount of heat lejected to the condensei and coolt'i 
is I b I L , wheie those quantities designate the total heat t j L b 
and at e respectively Fmthei, in the piocess cj of passing Llio 
expansion-valve theie is no change of total heal, by the pimciple 
pioved in Ait 72. Consequently, I f = I e . We may Ihcielou 1 stale 
the amount of heat reacted as I b I, 

Again, the work spent in the compiessoi is (in thcimal units) 
I b I a . It is the theimal equivalent of the aiea of the indicator 

b 

diagiam in fig 37, namely A VdP, which is equal to /,, 7 ft |>y 

a 

the geneial principle pioved in Ait 80. We are dealing licie wil h 
the mciease of total heat in adiabatic compression instead ol ils 
deciease in adiabatic expansion 

That these lesults are m agi cement with one another is seen by 
considenng the heat-account ot the cycle as a whole 

Woik spent = Heat icjected Heal Liken in 

1,,-Ia = (I* -If) ~ (la -I,) 

The coefficient of peifmmance, winch is the ratio ol the heat Uiken 
in from the cold body to the work spent m the compressoj , is 

la -If 
I*-Ia 

It will be obvious that the numerical value of this coefficient 
would be icduced if we weie to omit the cooling after condensation, 
which is repiesented by the line cle For in that case / would be 



iv] THEORY OF REFRIGERATION 145 

shifted Lo flic light, Lo a point on a line of constant total heat through 
d, and I t would be mci cased. The refrigerating effect would be 
lessened, but the woik spent in piodiicmg it would be the same 
as before, foi the indicator diagram of the compiession piocess, 
which is measured by / - J a , is not affected. The values of I a 
and l b depend only on the state of Lhe substance at a and at b 
icspcctively, and aic the same as before. 

113 Charts of Total Heat and Entropy for Substances 
used in the Vapour-Compression Process. The above results 
will show that calculations of peifoimance, as regaids refiigeratmg 
effect, heat icjected, and woik expended, become very easy when 
we can find the total heat of tho liquid just befoie the expansion 
valve and that of the vapour befoie and aftci compression This 
is leadily done if data aic available for chawing a Molhei chart of 
cntiopy and total heat loi the woiking substance Fanly complete 
elata arc available foi ammonia, cai bonic acid, and sulphuious acid 
Iff) charts foi these substances Mill be found in a Repoit of the 
Refngeialion Rcseaich Committee ol the Institution of Mechanical 
Engmecis ' . 

In dialing these chaits a gecmictiioal device is lesoitcel to for 
tlic [)urj)osc ol making the cliagiams at once open and compact, 
Milh tlie cflect that measuiemeiils ma} be made \\ith sullicient 
accmacv on a ehait of icasonable si/e This ele\ ice, which Molhei 
01 igmallv adopted in chauino lus 7r/> ehail toi cat borne acid, is to 
use. oblique cooiehnates, as illuslialcd in lig J2 The lines ot con- 
st, ml / aie hon/ouLil the lines ol constant </ nisteael of being 
pcipenelieulai to them aie inHiiud il a small angk The icsiilt is 
that ulien the chail is di.iun the eui\es on it aic shcaicd o\ci, 
.is compaicd with the I'oim lhc'\ Mould lake on a chait \\ith iccl- 
angular a\( s \\\v figuie \\liicli M r hcn diaun \\ith lectnngulai 
cooHluiales is i(lati\el\ Ie>u<> in e>uc diagonal elneelion may Mifh 
advantage be 1 opened out by the ILSC ol oblique cooidmales This 

* J\lui 1'iDi In^l Muh I'JiK/ , Uct I'lll Tlu i luu ts gi\ on Lhcu iiu chawn by 
Piolos^oi. ( ' K ,J( nkin Tho (halt foi (.uljunic IK id nnlinclic's icsulLs ot oxpcinnents 
hy MISSIH ilciikiii ,iiid Pj'u on tlio Llii'imal piopcilios ol thai suhNtaiico (Phil 
Tinn^ Roil tiu< , vnl i \ I'M), p (>7 and A 53J p 35}, which involve Homo collec- 
tion of an otulic i t liaii ])ul)Iiilu'd h\ r JJi IVrolhoi TIio data foi aintnonia arc those 
^ivoti by Alossirt (foocleiioiiyh and Moshoi (Bulletin No <>(> of tho University of 
llliiifiiH, I'Mtt) IMoiu iCLontly, conii)lolo liibloa oi Iho ihouiiodynanuc piopoitios of 
ammonia have boon ciitculalod with some\vhat diitoicnt numencal results by 
MoHSisKoyos and Riowtiloo (New Voik, John Wilov and Sona, 1910) In each of 
those publications a Molhtn /</i chaib is included 

u T 10 



146 



THERMODYNAMICS 



C'll 



is tme of I(f> charts, as applied to them, the deuce gives a beltei 
separation of lines that run moie or less diagonally acioss the sheet, 
like the lines in fig 31 (Ait. 101). Thcie is consequently a gu-al 
gam in clearness and in ihe power of accurately measmino I host 
changes of I that take place in lefngciatmg piocessos The in- 
clination selected foi the oblique axis will depend on I lie deyiee 
of opening out that is convenient in any particulai chart. In the 
case of fig. 42 it is 5 along the slope to 1 veilically, uid hence a 
measurement of I if made along a line of constant </> would have lo 




1 = 



Fjg 42 Use of obliquo coouhuilm in llio l</> < liul 

be interpreted on a scale five times as CCMISC- as UK nciniiil sc-.ilc- 
ioi /. 

An 1$ chart foi ammonia diawn wilh ohlujiu coouluLilcs is 
shown (on a small scale") in fig. d3 In llus c-asc llu- amounl ,,r 
sheaimg is moderate, foi the slope ot Ihe lines o! consla.U onliopy 
is only two to one. The dmgumi, foi Ihe useful .(<.,, ( , )n s,sls c,f 
a fan-like gioiip of hnes of constant pressure cvUii(lin M as si uuliL 
hnes though the legion oi wetness JVom Uu- JU,MK! I.(,.,,HL,,\M(. 
the vapour bounclaiy or saturation curve, and HUM, ,, s ( m ^ os mlo 

* For snrnlar charts i a m lulloi detail ami ou a male, I,,,,,,,, n <,wh f,u UH. ,u 
lems, reference should bo mado to tlu, 



IV] 



THEORY OF REFRIGERATION 



147 



the legion of superheat. Lines of constant temperatmc aie also 
drawn in the region of superheat, and lines of constant diyness 
(shown as broken lines in the chart) are diawn by dividing the 




43 /</) charL foi Ammonia 

sLi'dight poiLion of cacJi line of constant pressnie into a number 
of equal parLs Tins chai t should be compaied Avith that shoAvn for 
\\atcr and steam in fig 31 (Art. 101) m which, ho \vevei, there was 
no shearing, for lectangular cooichnates weie employed. Allowing 

102 



148 



THERMODYNAMICS 



[CM, 



i'oi that difference the leniarks made in Ail 101 apply lieu 1 . The 
slope of any constanl-pies.suic line, when piopcrly mteiprcled 
with reference to the coordinates used in (he. drawing, nieasuves 
the tempeiatuie, fbi T dljch/^ Ilcnc'e iheie is no abiupl cluium 1 
of dnection between tlic stiaii>h( pail of ciny such Jinc and ils 



/ 70 




41 !(/> cliait, fui (Jiulinnu An 



curved conciliations mlo Lhc liquid legion a I one end and into 
the legion ol'supeihcat al Uie olhei This ol eouise applies lo any 
substance. The /</., ehaiL foi siilj)luuoiis acid j,s ^eneially sinulaV 
to the chait for ammonia 

The Icj) chart foi caibonic acid is shown on a small s( ale in fi<> J, I-. 
It showt, the icj-ion ]ound abouL I he ciilienl point. Thai point 



IV] 



THEORY OF REFRIGERATION 



149 



coincides with the point of inflection of the continuous boundary 

curve ( Ai 1. 101 ). Constant-pi essure lines are drawn foi pi essures that 

are higher than the cutical pi essure as well as foi the wet region. 

The principle already stated applies to these lines, that the slope 

at any point (due regard being had to shearing) measures the tem- 

perature. In passing up along any line of constant pressure above 

the critical pi essure, the 

slope, which mcasui cs the tem- 

per atuie, mci cases continu- 

ously 1 . The stiaight poitions 

of the constant-pressure lines, 

within the boundary curve, aie 

divided by biokcn lines which 

aie lines of constant diyness 

Lines of constant Icmpeiature 

aic also diawn in the legion 

outside the boundary ctu\ e In 

thcicgion \\rlhm the boundary, 

where the state is that of a 

mixture of sal mated \apoui 

and liquid, these lines would of 

comsc be straight, and would 

coincide with lines of constant 

pic'ssiuc. To a\ oid confusion 

the sliaiyht portions ot the 

constaiit-tempcialurc linos aie 

omitted in the IIIHUC 

114 Applications of the 
I(f) Chart in studying the Va- 
pour-Compression Process) 
We aic now in a position to 
represent the vapour-comprcs- 

H! As Moaais .Jonkin and Pye h.uo pointed out (lou at , p 305) in eoirecting the 
u.ulmr c luut of MolJior, them is no point of inflection in any ol these lines For, since 

( (IT \ , winch is a positnc quantity throughout ihe whole 




r 45 Rofijgciation e\cle t-aced on. 
the /(/> chait foi Cailiunic Acid 



dl\ =T 
d<l>Ji> ' 



COUIHO of any lino above the cubical piosstno, as will he seen hy reference to the 
onliopy-temperatiuechagiam A point of inflection would icquire (^2) , to be zel 

Siuno of Lho constant jiiossuio lines \\cre ononeously di&vm \\ith inflections in 
Mollici's ougmal lef> cha?t foi caibonic acid, winch was lepioduced m the author's 
book on The Merhaniuil PindiiLtion of Cold 

| Parts ot this HI tide aio L,ikcn fiom an appendix (by the pieseut wiitei), to the 
Ropoifc of the Rofijgciation Kcacarch Committee of the Institution of Mechanical 
Engineers, 1914 



15 o THERMODYNAMICS [cir. 

sion refrigerating process by diagiams which exhibit the changes 
of total heat in relation to entiopy. With the help of Itf) charts 
numerical values of the total heat are readily found by measuic- 
ment at each stage m the assumed cycle. 

To tiace a lefngerating cycle on the appropriate chart, begin as 
before at a point a (fig. 45) which icprescnts the state of the sub- 
stance when it is about to enter the compiessor. This point is on 
the constant-piessure line corresponding to the process of evapora- 
tion in the cold body or evapoiatoi (fig. 36), and ils distance from 
the tovo boundaiy cuives coiresponds to the propoition of vapour 
to liquid in the mixtuie. If the compression is to be completely 
"diy," a will be on the boundaiy curve (at a^) more gencully tlic 
substance is slightly wet when compression begins. The stuight 
line (tb, diawn paiallel to the lines of constant entropy on the chart, 
is the piocess of adiabatic compiession The position of b is detci- 
mmedbythe mtei section of this line with a Jme ot constant prcssmc 
corresponding to the known upper limit of pressuie at which con- 
densation is to occur The temperature reached in the piocess of 
compiession is seen by the position of b among the lines of cons! an I 
tempeiatuie. In geneial there will be some superheating Bui il 
the mixtuie is so wet to begin with that the adiabatic line Ihiough 
a does not cioss the boundary cuive dining compiession bdoie 
the uppei limit of piessme is icached theie is none, and in Hint 
case the piocess is spoken of as " wet" eompiession. This would lie 
the case if compiession had begun at a c instead of a. J$y beginning 
at a it carries the substance into the legion of snpcihcat belorc 
compression is completed at b Next we have the constant-pi CSMH c 
process of cooling and condensation and fmlher cooling, icpic- 
sented in its thiee stages by the lines be, cd, and dc, the posilion of 
e being fixed by the tempcratme to which the liquid is known 
to be cooled befoie it reaches the expansion-valve. Then a hoi i- 
zontal stiaight line thiongh e (a line of constant total huil) 
repiesents the piocess of passing tlnough the cxpansion-vnh c , 
and determines a point/, on the evapoiation line, which exhibits 
the condition m which the substance enLcis the cvnpoiuloi. TIu 
process of evaporation Ja, which is the effective icfiigcinlmg 
process, completes the cycle The values of I a , 7 & , I c and I L (which 
is the same as I e ) aie lead directly by measurement from the chail 
As has been already pointed out, the woik spent in compiessmi> 
the substance is I b - I as and the icfugcrating effect is I a - I f 

We may illustrate the use of the chart by some examples. Take 
first a case in which the working substance is carbonic acid, with 



iv] THEORY OF REFRIGERATION 151 

10 C. as the temperalme of evaporation, 25 C. as the tem- 
peratme of condensation, and 15 C. as the tempera tuie to which 
the substance is cooled before passing the expansion-valve. The 
diagram for the perfoimance of an ideal machine undei these con- 
ditions is sketched m fig. 45, assuming various degiees oi dryness 
at the beginning of the compiession. If the substance is then 
entaely dry the operation staits at a lt namely, the end of the 
evaporation line for 10 C., and compiession brings it to b 
which is on a line of constant piessme equal to the piessme of 
satuialcd vapour at 25 C , namely, 930 pounds per sq inch. But 
the vapour is considciably supei heated at b lt its temperature there 
(as the lines of constant tempeiatuie show) being 58 C The woik 
spent in compression, Avhich is most accurately found by leading 
off the length of the line a^b-^ on a scale A\hich makes that length 
a dnect mcnsiue of the change of/, is 8 7 

We next tiace the piocess of condensing and cooling, under the 
constant picssmc of the condensei. Fiom b 1 to c the gas is losing 
ils supeihcnt, fiom c to d il is being condensed, and fiom d to e 
il is being cooled as a liquid. The point e is found b\ the intellection 
oi the line of constant piessme undci \\hich the piocess is cauicd 
out with the line of constant tcmpciature foi 15 C Nevt diaw 
f/paiallcl to the lines of constant total heat to meet thce\ apoialion 
line loi - 10 C The refiigualing effect / (li - /, is -179 The 
cocllicicnl of pcilbimance is theicfoie 5 5 This cycle con expends 
to completely diy compiession 

Suppose on the oLhci hand thai the compiession is just v\ct 
enough to dvoicl an\ supcihealmo In lhat case il musl commence 
lit a c in oulei Ihal llic adiabalic hue lepu'scntmg the compiession 
ma} pass lluough c on the hoimelaiN cm\e Then Ihe woik done 
in compiession is smallei than beioie 1 , foi a,c is smallei than a^b^, 
The icfngeiating efle-el is also smallei, foi fa, is smallei than Ja^. 
The c-oc'dicienl of peifoimance is now found to be 5 51- 

Bel ween these two thcie is a ceitain dcgice of diyncss which 
gives a slightly lughci cocfTicicnt of peifoimance than eithci This 
may be shown by laking a succession of poinls foi vanous stales 
of diyncss between a c and a as the staitmg point of the cj cle, and 
woilong out Ihc coefficient of peifoimance foi each But we may 
leach the same conclusion moic duectly as follows, by a general 
melhod which is applicable to any I(j> chait 

The icfrigcratmg effect foi anv slate of initial diyness, a, is 
proportional (on some scale) to the length ja. The \\ork done is 



152 



THERMODYNAMICS |cn. 



proportional (on anothei t,cale) to the leiu-th ah Hence the position 
of b which Avill give the highest coefficient ol' peifoimance is that 
which gives the smallest ratio of ab to fa This is found by diiwmfc a 
tangent from/to the line of constant picssnie ou winch b lus J?y 
applying this method the point b has been delcimmcd in the iigme, 
and hence the point a ib found at which compiession should bei>in 
if the coefficient of pcifoimancc is to have Us maximum value 
In the example that value is 5 72, and is obtained when llu 1 imhul 
diyness is about 87. 

As anothei example, still with caibonic acid, I tike I lie same con- 
ditions as befoie, except that the condensed liquid, instead of beino 
cooled at 15 C befoie expansion, i caches tin valve at I he lein- 
peiatuie of condensation, namely 25 C. In thai case Ihe ])roe(ss 
of expansion conesponds to Lhe line rlf,[ in fin. 15, [he lesl of I he 
cycle lemammg as befoie For maximum coclhcicnl ol peilormancc, 
unclei these conditions, compicssion should no lon^ei slail 1'ioiu a 
but fiom a point so chosen that Lhe ad in bade line llnough il leaches 
the constant-pi essuie cuivc b^c at the ponil wlieie the lani>c nl lioin 
J a meets that cuive This coiicsponds to an initial <Ii vness ol about 
95, and the maximum cocllicic.nl so obtained is [> ,'}<). When Ibis 
\ aluc is compared with that found in the pievious example, namely 
5 72, it \vill be obMOiis that a seiions loss ol cllicuiicv is eausul b\ 
omitting to cool the condensed liquid betoie it i caches Ihei \pansion- 
vah e 1 " 

A tuithei example "Kill seive to illnstiate I be application ol llu 
70 chait to caibonic acid \\oikmi> undei tiojiieal conditions, so 
that thelnghei limit of picssuic is alun e Ihe entical piessmeol llu 
subbtance Still taking- 10 (_' as Ihe teinpeuilnrc of <. vnpoi, >lion, 
we shall suppose the piessme in the condenser to lie 1200 pounds 
pei sq inch, and the tempeiahuc to winch llu liquid is eoolid 
befoie expansion to be 30 C 1 With these dala llu* diagiam I, ikes 
the foim shown in fig 1G, \\heic a^b^ lepiesenls a ])ioetss ol com- 
pletely diy completion, and ab a ])iocess of compression in \\hieh 
the position ot a has been <-o chosen as to i>ivt tlu- ma\imum co- 
efficient of peifoimancc The line ab consequently cuts Ihe euive 
of constant pi essuie foi 1200 pounds pei sq meh al I he place when 
a tangent fiom/ 1 would meet that euive. The point, r is delei mined 

*The nuuibeis givui in thosn ox.unplt^ A\CM^ louiiil l.y MKMIHIIICIIICHI, fioni 
' 



. 

Committee of the Institution otMK]i.mit,ill':n K iii( CMS wi'[iMis.ilinHt( ml llu-iiiiiiilifiH 
would be slightly diftcicnf but the gcnu.il u'sults would not l.o ufk-i (ml 



[V] 



THEORY OF REFRIGERATION 



153 



by following the curve of constant piessure till it cuts the 
line of temperatme foi 30 C The maximum coefficient of 
peifoiman.ee is obtained \b, 
when the diyness before 
compicssion is about 95. 
Its value is 3 1, and undei 
these conditions the va- 
pom is superheated to 
70 C al the end of com- 
piession The coefficient 
calculated foi completely 
dry compiession, when the 
compression line is fl 1 6 1 , has 
almost I he same value. 

In all these examples it 
is interest mo, anel practi- 
cally nnpoitant, to notice 
how little the coe lYicicnt ol 
pcifomuncc in the theo- 
ictical cycle is alfcctcd 
c\ en by considerable 
changes in the di \ncss 
be hue compicssion This 
is Line not onl\ of car- 
bonic acid but of any 
uoikmi> substance 

The application of the Mollicn tha^iam lo ammonia is illushated 
in lid -IJ, by an e\ani|)le iclciuny lo tiopieal conditions Theie, 
as in lonnei examples, UK lenipeiatiuc ol c\apoialion is taken as 
10 C 1 The substance is supposed to be condensed at 35 (' 
(piessmc 107-3 pounds pt'i stj inch) and to icmam at that tcm- 
peialuie' until it leaches the expansion-valve The cycle is abcdja 

Foi ammonia undei these conditions the follow mo icsults aie 
obtained by measuiement fiom the chayiam foi vauous A allies 
of the initial diyness 

Ihvnoss TtofiiKciatinff Woik of 

(Jompu'Hsioa 




Fi 4(i Ri hini'inliOM cyili 
\\ ilh Cuboni' Ai nl win n 
the uppi i limit (it pi i ">MIK 

1 i \u odi the LI jlii <i! [in SMiie 



Ttof updating 

Edoc'L 



Compii'ssiou 
I 
!)5 

85 
08 



201 
2 Hi 
2.J04 
2149 
199 3 



54 5 
WO 
454 
425 
398 



CoolThrient 

of 

Peifoimance 
480 
502 
507 
50(i 
r )01 



15 4 THERMODYNAMICS [LU. 

Here the maximum coefficient of pcifoimance is reached with 
a value of the initial dryness only veiy slightly gieatei than IhaL 
which just gives wet compression We may take the coefficient got 
by using wet compiession, with b on the boundaiy omvc, as 
piactically equal to that maximum. 




Fig 47 Rehigeiiitiou cycle with Ammonia uncloi Tiopiuil (JoudiUotiH 



As a final example, take ammonia wo i king in the same con- 
ditions as those that \\eie assumed foi carbonic acid in Ihc fust 
example, namely an evapoiation temperature of 10 C., a con- 
densation tcmpeiature of 25, and the liquid cooled lo 15 0. 



iv] THEORY OF REFRIGERATION 155 

before passing the valve. We then have these results for various 
values of bhe initial dryness: 



D iv ness 
befoie 
Compression 
10 


Refrigerating 
Effect, 
284 fi 


Work of 
Compression 

422 


Coefficient of 
Peiforraance 

674 


095 


2091 


384 


701 


09 


2535 


358 


708 


085 


2379 


336 


708 


08 


222 3 


315 


7 Ob 



The examples agiee in showing that there is vciy little differ- 
ence in the thcrmodynamic efficiency of the ideal peifoimance 
whcthei wet compiession is used or the initial diyness is adjusted 
to make the coefficient a maximum. 

The mle for this adjustment, applicable to any woikmg substance, 
may be cxpiessed thus Maik a point fon the evapoiation line to 
show the state of the substance on entering the evaporator. This 
point is found by drawing a line of constant total heat fiom the 
point which icprcsents the state of the substance when it reaches 
the expansion-valve Fiom the point/draw /& tangent to the line 
of constant picssuic foi the condcnsei, touching that line at b. 
Then I he compiession line ab passing through b is the one \\hich 
gives the maximum coefficient of peifoimance in the ideal cvcle 
Avith adiabcUic compicssion 

It does not iollow that the same dcgice of initial \\ctness Mould 
give I he maximum coclTicicnt in a ical compiessoi, loi the per- 
loim<mcc of a real machine is complicated b^ tiansfeis ot heat 
between the working substance and the metal In general such 
li.msfcis will be less when the working substance is dry On the 
olhci h.md, with a wet mixUiic, what is called the volumetnc 
eflicic-Mcy of Hie apparatus is gieatcr, since a laigei quantity of 
I he working substance passes through the machine lor e\ery cubic 
fool swept through by the piston, and this tends to reduce the 
proportion of Ihose losses that arise from mechanical friction, 
and from radiation and conduction between the apparatus and 
rts environment 

115 Vapour-Compression by means of a Jet. Water- Vapour 
Machine Whatever be the working substance, an essential feature 
of any vapoui-comprcssion refrigerating machine is that the vapour 
must be pumped up from the low-pressure legion in which it has 



156 THERMODYNAMICS [cir. 

been evapoiated to the high-pressuie legion in which it is to be 
condensed. But this pumping up may be effected in move than 
one way The usual way is by means of a cylmdei and piston, and 
so long as the vapoui-piessme is modeiately high Lhe use of a 
compiessmg piston is quite satisfactoiy. But when the vapoiu- 
pressuie is veiv low, as it would be if water weie used fov ihc 
vvoikmg substance, the volume to be swept thiougli bv a com- 
pressing piston would be so laige as to be veiy inconvenient, and 
the amount of \\oik ^hich Mould be wasted thiougli fuel ion 
between the piston and cylmdei would be an excessive addition 
to the legitimate woik of compiession. Not only would Lhe machine 
be exceedingly bulkv but its practical efficienc}' -uould be exceed- 
ingly low At C , foi example, the density of waler-vapom is 
so small that about 3G5 cubic feet of it aie rcqiined Lo absoib as 
much latent heat as one cubic foot of ammonia vapour Hence to 
use watei-yapom as a lefngeiatmg agent some appliance must be 
icsoited to which will avoid the bulk and fnclional waste of an 
oidmaiy compiession pump One such appliance is a ccntiifuoal 
pump 01 icveised tin bine anothci is an ejector or jet pump, in 
which an auxihaiy stieam of vapom, supplied at a compaialivcly 
high piessme, Ibims a moti\c jet which drags with it the \.ipom 
to be "aspnated," namely the vapom which has been foinud bv 
evapoiation at low picssuie, so that both pass on tooolhci lo bi 
condensed This device is applicable to any ihucl, and closal- 
ciicmt sj'stems \\hich operate on this punciple ha\e lx.cn devised 
Foi othei woikmg substances besides u atci-\ apour The vajwiu of 
the motive jet necessauly mixes with I he \apoiu Lo Ix cispnalcd 
and both aie condensed togethei theie aie thus two cncuils \\hu-li 
coalesce in the condenser. PaiL of the condensed liquid reluius 
thiough the expansion-valve Lo the cold cvapoinLoi, and ,u-ls as 
the effective woikmg substance m producing icfimei.ilion llu 
other pait is foiced by a feed-pump into a boilci where iL is vnpo. .ml 
at a lelatively high piessuie, so that iL ma> ac-t as Ihc motive |d 
the t^o then meet again in the ejector on their way Lo (he c'-on- 
densei. 

It is however when watei is the woikmg substance Llul such a 
system is specially applicable An independent supply of |, O ,I CT 
steam foims the motive jet It acquncs a high veloc.ty , passmo 
thiough a discharge nozzle, which comcigcs lo a place al which 
the low-piessme vapom to be aspuated is allowed access The Inh 
velocity jet communicates pait of its momcntmn lo lhal vapour 



iv] THEORY OF REFRIGERATION 157 

and the two pass on in one stieara to the condenser through a 
divergent pipe in which the stream loses veloci ty and gains pressure 
as it pioceeds. This enables the pressuie of the woiking substance 
to use from the lo\\er to the upper of the limits between which the 
machine works, namely fiom the low piessme at which the 
aspirated vapour is foimcd to the higher piessme at which it is 
condensed In lefrigciatmg machines constiucted to act in this 
way the quantity of vapour in the motive jet is as much as three 
01 even five times I he quantity that is aspirated The thei mo- 
dynamic efficiency of the method is found on tual to be only 
moderate, but the apparatus has advantages m point of simplicity, 
and in the absence of any Avorking substance other than watei 
It has been applied not only to cool water, but also to maintain 
a tcmpeuituie considerably below C , in which case bime is 
substituted for fiesh watei as the woiking substance whose vapoui 
is aspuated, and the cooled bune is pi evented fiom becoming too 
dense by systematically letuimng to it a quantity ol watei to 
make good Ihe amount that is c\apoiatcd 

116 The Step-down in Temperature Use of an Expansion 
Cylinder in Machines using Air So long as the woiking sub- 
stance in a relngeiating machine is a \apoui -\\hich becomes 
liquified dining Ihe opuation, it is piacticable, as \\e luuc seen, to 
dispense with an expansion cylmclci and still have a laigc amount 
ol iefii<>cialing elleet The slep-down in tempciatinc, uhich is 
nceessai} in any lefngciation cycle, occms as a, consequence of 
Ihe piocess of thioltlmg, while the subslancc passes the expansion- 
valve Tins is tun also of a gas neat its eiilical point, and hence 
a machine using eaiboiuc acid undci Uopical conditions can be 
ffectivc without an expansion c\lindci all hough the substance 
may not undeigo liquefaction A gas noai its culical point is \eiy 
lai fiom pci I eel and does not even appioximatcly confoim to 
Joule's Law A gas nluch conlonns to that law would suffei no 
-.Icp-down of lenipciaLuie in passing an cxpansion-vah e (Ait 19) 
With a gas such as an, which is neaily pcifcct at the tempeiatmcs 
ind prcssuies that occur in oidmaiy icfiigciation, the step-do vui 
uould be too small to seivc the clesned puiposc Hence with an 
P.OI woiking subslancc an expansion cyhndci becomes an essential 
ic'incnt of the machine Rcfiigeiatmg machines using air, and 
20olmg it by means of expansion in a cylmdei in which it does 
work against a piston, are amongst the oldest effective means of 



158 THERMODYNAMICS [en. 

pioducing cold by mechanical agency. They arc still used foi the 
diiect cooling of the atmosphcie of cold stoics, but their use is 
now less common, because machines in which the woikmg substance 
is a condensable vapom aie not only moie compact but give a better 
theimodynamic return for the woik spent in dining them. 

117. Air-Machines. Joule's Air-Engme reversed. The au- 
machmes which aie in actual use opeiate by taking in a poition 
of air from the chamber that is to be kept cold, compiessmg it 
moie or less adiabatically with the result that its tempcialure 
uses considerably above that of the available water -suppby, then 
extracting heat fiom it in the compiessed state by means of cu- 
culating watei, then expanding it in a cylinder in which iL docs 
work, with the lesult that its initial piessure is icstoied and its 




Cuu'c.1 A 
Fig 48 Oigans of an Aii-Mdcliinc 



temperatuie falls gieatly below the initial temperatme. It is Lhcn 
leturned into the atmospheie of the cold chambci, with which 
it mixes, the object being eithei to lowei the tempciatuic in the 
chambei 01 to keep it fiom using thiough leakage ol heat iiom 
outride This type is known as the Bell-Coleman an -machine. 

The cycle is a icversal of that of Joule's Aii-Engme, descubcd 
in Ait. 5-1. As applied in lefngeiation the appaiatus takes the 
foim shown diagrammatically in fig. 4S In the phat>e of action 
shown theie the pistons are moving towaids the left Air fiom the 
cold chamber C is being diawn into the compiession cylmdei M. 
In the retuin stioke it will be compiessed fiom one atmosphcie 
to about fom, with the result that its temperature may be uuscd 
to 130 C. or higher. It is dehveicd under this piessuic to the 




tv] THEORY OF REFRIGERATION 159 

coolei A wheic it gives up heat to the en dilating water and comes 
clown to near atmospheric tempera tin e It then passes, still at 
high pressure, to the expan- ,, 
sion cylinder N, wheie it does 
woik in expanding to the 
initial pressure of one atmo- 
spheic and theieby becomes 
very cold, i caching a tem- 
perature of peihaps 60 C. 
or 70 C , in which condition f 
it is ictmncd to the cold 
ehambei An ideal indicator v 

diagram for the whole cycle Fl S 49 Inihcaloi Dingiam of Aii Machine 
is gn en in fig 10, where fcbe shows the action of the compression 
cylinder and cadf shows that of the expansion cyhndei The aiea 
abed mcasuics the net amounted' work that is expended In the 
diagiam the compiession and expansion aic both tieated as adiabatic 
and the \ olume ol A as well as that of C is assumed to be so laige 
that dining delivery of Ihc an its prcssme does not sensibly change 
Willing T a , T,,, T L and T tl i'oi the tempeiatme of the vioiking an, 
at the points a, b, c and d of the chagiam, we ha\ c Q , = A p (T b -T n ) 
(bi the heat icjccted to the cooling watei, and Q, = K v ( r l\ - T d ) 
for the heat usclully extiactcd liom the cold ehambei The net 
Limounl ol \\oik expended is equal to Q.i Q.t The coefficient of 
peiloimanec is Q 

Q.-Q, 

Foi the icasou explained in Ait. 54' 

T T T T T 

T "=' a , fiom which ^' T ' l = ' 

J c l d * l> ~~ * a * b 

TT, ,, Q < ~ T " 1 q " T <- 

Jlcnce ,. - ,., , anci ,. r ., , . 

' e i J b ( t i V< J- T, J- o 

This coelFieicnt of pcifoimance is low because of the M.IY laige 
iniige ol tcinpeiature Ihrough which the woiking air is earned 
For llns leason, and also because of gicatei fnotional losses, an 
Liclual an -machine gn r cs lesults that compaie unfavouiably with 
those obtained in the vaponr-compiession piocess. 

Consideicd as a means of pumping up heat fiom T c the tcm- 
pcratuie of the cold ehambei fiom which heat it, taken in, lo T a 
the tcinpciatine of the cnculatmg water to which heat is dischaiged 
the aii'-macliinc has two serious theimodynamic defects Theie is 



160 THERMODYNAMICS |c. 

an llle^ eisible tiansfei of heat when the woiking air, aftei bung 
heated by compiession to 2\ comes into thermal contact vuth the 
cnculatmg watei at T a . and theic is another ineveisible tiansl'ei 
when the vsoikmg an, chilled by expansion to T lt , mixes with Lhe 
less cold almospheic oi' the chamber aL f l\ An ideally efficient 
icfiigeiatmg machine, namely a ie~\erscd Cainot engine, working 
between T u and T ( as uppei and lowci limits \\ould ha\ r c (Ail 105) 
a coefficient of pertoimance equal to 



The coefficient L'onnd abo\ e foi the ie\eiscd Joule cycle is sub- 
stantially less, because T b is highei than T a . 

In the piactical woiking of such machines the piesenee of 
moistme 111 the an has to be icckoned with The air coming from 
the cold chamber is more 01 less^saturated dining expansion it 
becomes super-satin ated and the water fiom it would IDC deposited 
as snow in the expansion cyhndei, and might mteifeie wilh Ihc 
action of the mechanism, it pievenLive de\ ices weie not mtio- 
dnccd One such device is to divide the whole expansion into U\o 
stages by making it compound In the fust stage the expansion 
is caiiiccl only far enough Lo cool the an to atemperatmc just above 
the freezing point In that way neail^ all the moist me is deposited 
in the foim ol watci, and is easil} diaincd away befoic the final 
stage, winch would ficeze it, begins. Anothei de^ ice is to condense 
out most of the moist me befoie expansion, b} passing the com- 
piessed an thiough pipes \\hich bimg its tempeiatuie down to 
near the fieezmg point befoie it enteis the expansion cylmdei. 
These '"cliyiiig pipes," aie kept cold by an iiom the cold chambei 
that an is consequently warmed by them, but the loss is made good 
by the lower lempciatme which the working an i caches in ex- 
pansion, as a consequence of the piecoolmg it has undergone in 
the drying pipes. 

118 Direct Application of Heat to produce Cold. Absorption 
Machines In another class of icfiigeiatmg appliances theic 
is no application of mechanical powei the agent is heal, winch is 
supplied fiom a high-tcmpeiatuie source, and is employed in such 
a AS ay as to cause another quantity of heat to pass fiom a cold body 
and to be discharged at a tempeiatuie mtei mediate between that 
of the cold body and the hot souice In such machines the clli- 
ciency of the action fiom the theimoclynamic point of view is 



] THEORY OF REFRIGERATION 161 

easuied by the heat latio -~ where Q 2 is the heat extracted 

Ml! 

:>m the cold body, and Q is the high-tempeiature heat which is 
pphed to cany out Lhe opeiation. 

A typical example is the ammoma-absoiption refngeiating 
achnie Essentially this is a device in which the vapour of 
nmonia is allemalely dissolved by cold watei under a lelatively 
\v picssuie, and distilled fiom solution in water under a lelatively 
gh piessuie by the action of heat The ammonia vapour, driven 
f by applying heat to a solution is condensed m a vessel which 
kepi cool by means of en dilating watei. This gives anhydrous 
[incl ammonia at high piessure which (just as m a compiession 
achme) is allowed to pass thiough an expansion-valve, into a 
il or vessel foimmg the evaporatoi. A low prcssuie is maintained 
the evapoialoi by causing the evapoiatccl vapoui to pass into 
other vessel, called the absoiber, wheie it comes into contact 
Lh cold watei in winch it becomes dissolved. When the water 
the absoiber has taken up a sufficient piopoiLion of ammonia 
in tin n is heated to give of! Lhe vapoui agcTin unclci high piessuie 
the simplest foim of Lhe appaiatus the same vessel selves 
ematcly as absoibci and as genciatoi 01 distillci Foi con- 
uious woikmg thcie aie sepaiate vessels, and the iicli solution 
tiansfeiicd fiom the absoibei to the general 01 by a small pump, 
nlc the watci fiom which ammonia has been expelled (Ions back 
the absoibci to dissohe moie ammonia The scheme of such an 
pai a lus is shown m iig 50 Ilcal is applied to (lie solution in 
e ociicidtoi bv means ol a steam-coil. The <>as passes off at top 
Lhc coudcnsei, then thiouqh tlic e\pansion-\alve to the e^apol- 
>i, and I hen on to Lhc absoibci, wheie it meets a cmient of watu 
vciy weak solution that has come ovei fiom the bottom of Lhc 
ncniloi BelAvccn the gcncialoi and absoibci is the mtcichangei, 
levicc foi cconomi/mg heat by taking it from the watei that is 
Unnmg to the absoibei, and giving it to the rich solution that 
being pumped into the gcneiatoi This nch solution is dehvcicd 
Lhc top o( the column in the gcnciator, as the liquid paits \\ith 
e ammonia it becomes denser and falls to the bottom wheic it 
:apes to the absoiber thiough an adjustable valve When water 
soibs ammonia a laigc ninount of heat is given out Hence the 
sorbei as well as the condenser has to be kept cool by means of 
culatmg water or otherwise. Under the most favourable cou- 
pons the quantity of heat which such a machine takes in fiom 



162 



THERMODYNAMICS 



[en. 



the cold body is considerably le^s than tlie quanliLy ol' high Iciu- 
peratuie heat that has to be supplied, for it needs nioie theimal 
units to separate ammonia gas from solution in watei than simply 
to evaporate the same amount of liquid ammonia 

In another type of absoiption machine watei-vapoui is I he- sub- 
stance which is absoibcd it is taken up by sulphinic acid, fioni 
which it may again be sepaiated by the agency of heal Such a 
machine has been used for ice-making, the evaporation of part 
of the watei servino to fiecze the lest In this case also the heat 



Steam\ 




Retjulating Value 
Fig 50 Oigans of an Ammonia Absoiption M.uhiiR- 

ratio, namely the latio of heat usefully extiacted to heat supplied, 
is less than unity, for it takes moie heat to separate the vapom 
of watei fioni a sulphuric-acid solution than fiom pine walci It 
is a familiar fact that when watei is mixed with sulphuric acid 
much heat is given out 

It is obvious that a better thermodynamic result would be 
attainable if the piocess of absoiption of the vapour AVCIC attended 
by the giving out of less heat than is equivalent to the latent heat 
of the vapour itself. This is the case when ammonia vapoui unites 
with ceitain anhydrous salts, for which it has much affinity, such 
as the sulphocyamde of ammonium (NII 4 CNS), 01 the nitrate, 
biomide or iodide. Any one of these salts forms a suitable absoi- 



] THEORY OF REFRIGERATION 163 

nt. The ammonia vapour unites with the dry salt to form a 

[uid solution, fiora which the ammonia vapour can again be j 

ivcn oif by the application of heat, leaving the salt dry and i 

ady to serve again as the absorbent The vapour is stnctly 

ih vdi ous, lor no watei is present in the woikmg substance at any 

igc. The heat gi\ en out dining absorption of the ammonia j 

ipour by the salt is substantially less than the latent heat of 

e vapour itself at the same picssuie, foi pait is taken up in j] 

Kiefymg the salt. Similaily the heat icquned to effect ascpaiation ' 

ammonia vapour from the salt is substantially less than the j 

tent heat of the vapoui, foi part is supplied by the solidification 

the salt Consequently, when this piocess is made use of foi the , 

n pose of lefngeiation, the latio of the heat which is exti acted ' 

Din the cold body to the high-tempeiature heat, which is supplied ! 

Lhe gcnciatoi, would be gieatei than unity, if it weie not foi 
ch losses as occui through impeifcction in the working f 

This piocess is the subject of patents by Mi W W Seay*. In 
s icfngeratmg appaiatus the woikmg substance is made up of 
)out 8 Ib of the salt to one of anhydious ammonia. Theie aie 
io (or moie) similai vessels each of \\hich seivcs alternately as j 

)soibcr and as genciator These aie cylinder A\lnch aie kept 
)\vly icvolving as a means of slimng the nuvtuie Piecau- 
>ns have to be taken, b> selecting a suitable matenal ioi Lhe 
'ssels 01 foi then lining, to avoid chemical action on the pait of 
ie salt In each vessel llieic is a eoil of pipe thiough \\Iuch cold 
-ilei ciiculatcs while I he \ essel is aclmy as absoibci, and hot watei 

steam while it is acting as gcneialoi The othei oigans aie Lhe 
me as in anv olhei compulsion 01 absoiption plant The 
nnionid \apoui passes J'lum I he gencialoi to a suilace condense i 
lieicits lalcnt heat is dischaiged to circulating watci, Ihentlnough 
i expansion-valve to Lhe e\apora,toi, where it takes up heat fiom 
e bime 01 othei body that is to be coolcel, and then passes on 

Lhe absoibei In the geneialoi anel condenser its picssuie is 
latively high in the cvapoialor and absorbei it is low 
Tesls of a heay machine show that, e\cn in small sizes, theie is 
much greatci amount of lefiigeiatmg effect foi the same expen- 
tuie of heat than is found in machines which work by the absoip- 
ni of ammonia in watci As applied to ice-making it appeals that 
e Seay machine will produce as much ice, per Ib of coal consumed, 

* BuUsli Patent (Marks), JSTo 25806 of 1907. 

112 



164 THERMODYNAMICS L cn - 

as can be obtained by employing a good steam-engine to duve a 
good vapoui-compiession lefngeiatmg machine. 

119. Limit of Efficiency in the Use of High-temperature 
Heat to Produce Cold Any appliance, such as an absorption 
machine, foi the chiect production of cold by the agcncv of heat, 
icqunes a supply of heat at a tempeiatuie highei than that of the 
suiioundmgs. There aie nccessanlv tlncc tcmpeiatiucs lo be 
consideied. (1) the low tempeiatuie T 2 of the cold body fiom which 
heat is being extiacted, (2) the mteimediatc tempeiabuie I\ of Lhe 
available condensing watei 01 othei "sink" into which heat can 
be i ejected, and (3) the high tempeiatuie T of the somce from 
which heat is supplied to perfoim the operation Any such ap- 




Moto 



-j- ^i and ^ - 

Q | Refrigerator 2 



Fig 51 

pliance may be regaided as equivalent to the combination of a 
motoi 01 heat-engine dnvmg a lefngeiator 01 heat-pump (fii> 51) 
A quantity Q of high-tempeiatuie heat goes in al one place, and 
theieby causes a quantity Q 2 f low-tempeiatuic heat lo go in at 
anothei place Heat is i ejected at the mtei mediate icinpciaLuie 
T 15 and the heat so rejected is equal to the sum of Q and Q 2 , foi no 
woik is done by the appliance 01 spent upon it, as a whole This 
descuption applies \vhethei the appliance is actually a mechanical 
combination of a heat-engine with a heat-pump, or is an absoiption 
machine with no com eision of heat into "uork and woik into heat. 
In cithei case we have to considei what is the ideally gicatcst lafoo 
of the low temperature heat Q 2> which is extiacted fiom the cold 
body, to the high- temperature 01 driving heat Q, when the thice 
Icmpeiatuies T z , 2\, and T are assigned. 



'] THEORY OF REFRIGERATION 165 

Suppose, first, thai the machine consists of a peifect (icvcrsible) 
iat-engrne duvrng a perfect (reveisible) heat-pump. Then it is 
isy to calculate the lalio of the heat extracted Q 2 to the heat 
ipphed Q Wilting W foi the heaL-equivalent of the work de- 
'loped in the heat-engine and employed to drive the heat-pump, 

e have (by Art. 38) HIT T \ 

w " ^ i' 
" ~ T 

ice the heat-engme is reveisible. Again, since the heat-pump is 

so reversible, n IT T \ 

rr/_ ( &U i - ^2) 

T 2 
.Art. 100 Hence = -_) 



Inch gives the required ratio of heats 

The impoitance of this result lies in the fact that no other 
cthod of applying heat to produce cold can give a higher ratio 
Q 2 Lo Q, the Llnee tcmperaLures T, r l\ and T 2 being assigned. 
) pio\e this, imagine the combination of reversible heat-engine 
id reversrble heat-pump to be reversed rt will then gi\e out an 
nount of heat equal to Q to Ihe liot body and an amount equal 

Q 2 to the cold body, and iL will take in an amount equal to 
+ Q horn the rnLcimcdiale boch at r l\ It Mill still develop 
> \\ork as a \\hole, noi ie<iimc \unk to be spent in dining it. 
lagme liutlrei thai between the hot bod> and the cold one theic 
e two appliances vujikmg both using the same intermediate 
mpcialuie one ol which is this icveiscd combination and the 
hei is a Kliigc'ialiug machine (such as an absoi phon machine) 
lose elhcKncy \\e wish to compare wilh lhal ol Ihe combination. 
u-n il it \\cie possible for that machine to ha\ c a higher enicicncy 
an Ihe combmalioii, it uould exliacl more heat than Q 2 fiom 
e cold body for Ihe same expenditure ol high-lcmpcialure heal 

lie nee, when both work together, namely the combination 
Diking i excised and the othci machine working direct lire cold 
idy \\oulcl lose heat while on the whole the hoi body would lose 
lie In olhei woids we should then have an impossible result, 
nicly a simple transfer oi heal, by a purely bclf-aetmg agency, 
>rn Ihe cold body al 7' 2 lo a wanner body at J\, the intcr- 
scliate tcmperalmc. The agency would be self-acting in the sense 
being actuated by no form of energy, mechanical or thermal 
ich a result would be a violation of the Second Law (Art 31) 
le conclusion is that no means ol employing heat to produce cold, 



]66 THERMODYNAMICS [en 

whether dncctly, as in .in absoiption machine, or milnocLly as in 
a compiession machine dm en by an engine, CMII be moio efficient 
(foi the same thiee tempeiatuics) than the combination of a 
reveisible heat-engine dining a icveisiblc heat-pump Hence the 
expi ession q^ T 2 (T1\] 



measuies the ideall}' gieatest latio of heat ex Lidded, to heal 
supplied. Any real appliance will show a smaller heat ralio in 
consequence of irreveisible featuies in its action The action of 
an ammoma-and-water absoiption machine, for example, is very 
far fiom being leveisihle the heat ratio in iL is much less Ihan 
unit} 7 -. But, as the above expi ession shows, Avlicn T, is no I much 
lower than T 1 and T is much highei, Q> may be much greater Ihan 
Q in the ideal use of heat to produce cold 

120. Expression in Terms of the Entropy The above ex- 
pi ession foi the ideal perfoimance under reveisible conditions may 
be written Q (\ 

H_ , rr\ rr\ \ ML2 / rn m \ 



f 11 

fiom which --7ir T 

1 2 

This expiesses the conseivation of cnliopv lf>i HIP complc-le 
ie\eisible opeiation The entiopy of the system as u wliolc does 
not change. Foi the teim on the left is the gam oi entropy by Ihe 
body at T t to which heat is i ejected the l\\o leims on Ihe nghL 
are the losses of entiopy by I he hot bod} and cold body icspcc lively 
The whole action may be legarrled as a tiansfei ol entiopy fiom 
two somces at T and T 2 to an in bei mediate sink <il 7\. So long as 
the action is leveisible this tiansfei occms without affecting the 
aggiegate entiopy, but if it is not completely reversible the ag- 
giegate entropy will increase m that case the teim on Ihe lei I 
becomes gicater than the sum of the teims on UK n^lil. 

Again, the equation shows that, under rcvcisible eondilions, Ihe 
pioductof the eutiopy lost by the hot somce (Ihiough I lie lemoval 
of the heat Q) into the diop in tempciatmc Avhich that heul undei- 
goes, namely fiom T to T lt is equal to the pioduel of Lhe entiopy 
lost by the cold body into the use of tempcuitme of the abstracted 
heat Q 2 . Each of these pioducts is in fact a measure of W, the woik 
which the heat-engine pioduces, and the heat-pump consumes, m 
the ideal combination of reversible engine with icvci&iblc pump. 



'1 THEORY OF REFRIGERATION 167 

A mechanical analogue is illnslialed in fig 5'2 Heie a quantity 
' vvatei M, supplied at a high Jevel H, descends to a lower level 
! and serves to raise another quantity M z fiom a still lower level 
2 up to II \ Both quantities are discharged at the level H r The 
relation is reversible, and the eneigy equation may be written 

M(TI- 7/j) = A/ 2 (7/j - //,) 




/M 
H 




Fin r>2 Miclianinil imAlojriio of I hf usr ol lic.it lo proilncc' cold 

hi com pa nng I his wilh Lhc equation gi\ en above, i'oi a coiresponcl- 
i" reversible Iheimalopciation, it will be noticed that the analogue 
f weight (ol water) is not heat but entiopy, namely the quantity 
I' heat divided by Lhe tcmpcifituic ol' siipplv. 
The levcisible llieimal opeiation may be lepicscnted on the 
nli'opy-tempciatuic diagiam as m (ig 53 There the area abon 
L'prcsciiLs the high-temperature heat which is supplied at tcm- 
cvatuic T, and the area fibcd icpic&cnts the work which would 



168 THERMODYNAMICS [en. 

be done in a peifect heab-engme by letting clown that quantily 
oi' heat from T to the lower level 2\ Between the given levels of 
temperatuie T a and T 2 diaw a icctangle dcfg whose aiea is equal to 
the aiea abed, and pioduce cf to meet the base line lor zeio tcm- 
peiatuie m m. Then the area fgnm icpiescnts the refugeiatmg 
effect, namely the heat exti acted fiom the cold body at 2' 2 ' The 




m no 

Fig 53 

amount of heat dischaigccl at the in tei media k 1 level T^ is equal lo 
the aiea ecom, which is equal to the sum of the aieas abon and 
Jgnm 

121 The Refrigerating Machine as a means of Warming. 
In any such appliance, whether levcrsible or not, the quantity of 
heat delivered at the intermediate tempeiatuie 'J\ is greater than 
the quantity supplied at T by the amount of the heal uiisccl fiom 
T 2 , and may, as ue ha^ c seen, be much giealei This fact is the 
basis of an mteicsting suggestion made b} Kelvin in 1S52, that 
in Ihe warming oi looms it would be Iheimally moic economical 
to apply the heat got fiom binning coal in this indirect way than 
to discharge it into the room to be waimcd. The thcimody mimic 
\ alue of high-tempeiatine heat is wasted if we allow it dncetly to 
entei a comparatively eold substance That value might be better 
utilized by employing the heat to pump up moie heat, taken in 
horn say the outside atmospheie, to the level lo which the room is 



1 THEORY OF REFRIGERATION 169 

be warmed By using, foi example, an efficient steam-engine 

chive an efficient heat-pump, a small quantity of heat supplied 
a high tempeialure will suffice to laise a much gieatei quantity 
heat thiough the small range that is requned, and consequently 

produce a much gi eater warming ellect. Similarly, if a supply 
' power fiom any souice is available as a means oi warming to 
moderate tcmpeiatme, it will be tinned to bcttei account if we 
t it to duve a heat-pump than if we simply conveit it into heat, 
he suggestion that some of the coal which is used loi heating 
>oms might be saved by applying heat in this indnect mannei 
is at piesent no more than a theoietical mtciest. 

122. The Attainment of Very Low Temperature, Cascade 
lethod Anothei pait oi the science oi lefngeiation deals with 
icthods of pioducing cold so exticme as to hqueiy an and other 
^-called pcimanc-nt gases This is now the basis oi an impoitant 
iclushy, which employs the liquefaction of an as a step to ^\ a ids 
ic sepaiation of its constituents, with the object oi making com- 
icicial use oi the oxygen 01 the mtiogen or both To liquefy 
ny gas the lempeialme must be icduced below the cntical point 
Vit. 77), and foi mtiogen this moans a cooling below 110 C 
'empeialuics much lowci than thi^ have been i cached by the 
K' I hods \\luch \\ill now be descnbcd Hjdiogcn, whose cntical 
:mpeiatine is 211 C , has not only been hquehccl but M>hdilicd 
s melting [joint unelei atmospheiic piessnic is about 25S 3 l. 
i 15 absolute E\ en helium, tlie most leiiactoij ol all known 
ases, has been liquefied nuclei conditions that lo\\ucd the' 
cmpcialinc to within I luce or i'oiu dcgiccs oi the absolute /eio 

One uay oi leaching a VIM low lempeialme, called the "cascade" 
iicthod, is to have a seues oi compicssion uJngeiating machines 
o connccUd that Ihc \\oikmg substance in one, when cooled by 
ts own cvapoiatiou, acts as the cnculatmg fluid lo cool Ihc 
ondcnsc'i oi the next machine ol the SULC.S, and so on Difleient 
\ r oikmg (luids aie selected loi the successive machines, so that 
ach in tin n ic aches a lowci lempciatnie than its picdccessoi 
The gcncial idea oi the method is illustiatcd in iig. 5J In that 
hagiam the lust woiking substance is caibomc acid, which is 
cpicsentcd in the sketch as supplied from a icservoii on the left, 
nlo which it has been compressed It expands thiough atlnottle- 
iilvc into the vessel A, fiom which it escapes at atmospheiic 
nessuie (this pait of the apparatus might be completed by a 



170 



THERMODYNAMICS 



CTT 



compression pump restonno the substance Lo the icscrvoir). The 
effect is that the vessel A is kept at a temperatuic of abouL - 80 (' 
Within it is anothei vessel which ser\ r es as the condenser of d 
machine using ethylene as working substance Ethylcnc has a 
cutical tejnpcrature of 10 C., and needs only a modciatc 
piessnie to liquefy it at 80 C. It is pumped into the inner pait 
of the condensei A, is thcie liquefied, and passes on through an 
expansion-valve to the outer part of the vessel B in which if eva- 
porates, pioclucmg a tempeiatme of say 130 C al the low 
pressure which is maintained by the pump. Thus cools I he vessel 
B below the critical point of ox}'gen (namely 118 C ), accoid- 

Ethylene Oxygen 




-80C -130 C -200 C 

Fig 54 Cascade Method of leaching veiy Low Tempeiatmoq 

mgly oxygen may be used as the woikmg substance of I he next 
machine It is condensed m the mnei part of the condenser Ji, 
and aftei passing thiough an expansion-valve it may pioducc a 
tenipciaturc of - 200 C or less in the vessel C by cvapoialmg 
theie nndci alowpiessmc Each machine of the sencs is a vaponr- 
compression machine, vioikmg on the pimciple already dcsciibed, 
and made up of an evapoiatoi, a compicssmg pump, a condensei, 
and an expansion- valve The essential featuie m the combmahon 
is that the woikmg substance in any one machine must be e\a- 
porated at a tempeiatme that is lower than the ciitical poml of 
the woikmg substance of the next machine m the scucs. 



v] THEORY OF REFRIGERATION 171 

123. Regenerative Method Bui it is in a dificicnt way that 
ow tcmpciatmes arc now attained for the commercial liquefaction 
f an. The usual pioccss is a regenerative one, first successfully 
levcloped by Lmde, in which the Joule-Thomson effect of nrever- 
ible expansion in passing a constncted onfice serves as the step- 
lown in tempera tin c, and a cumulative cooling is produced by 
ansing the gas which has suffered this step-down, to take up 
leat in a thermal inteichangei from anothei poition of gas that 
s on its way to the ounce. 

Consider first what would happen if theie weie no such theimal 
nterchange. Imagine a gas such as an to have been compiessed 

a high prcssuic P i, and to have had the heat developed by com- 
nession icmoved by circulating watei 01 othenvise, so that its 
empeiatuie is that of the suuoundmgs. Call this initial tem- 
)eiatuic TI Let the compiessed gas at that tempcratiue enter an 
ippaiatus in whicli it expands irreveisibly (thiough an expansion- 
'alve 01 plug or constricted ounce of any kind) to a much lowei 
HCSSIIIC Ph, at which prcssuic it lca\ es the appaiatus If the gas 
vcie an ideal peilccL gas this nie\ cisiblc expansion would cause no 
all in tempciatiuc In a ical g tl s Iheie is in gencial a fall, fiom 
C l to some lowci tempciatuic T. The fall 2\ - T mcasmes the 
Foule-Thomson cooling effect of the gi\ en chop in picssuie. In 
Foiile and Thomson's cxpciiments on an it was about a quailei 
)l a clegue foi each atmosphcic of chop in picssuu ! 

Tlic cooling c fleet of the chop in picssmc may he measuied by 
he quantity oi heat which uould \\&\ c lo be supplied to the gas, 
)ei Ib , afici expansion, to icstoit it to the tcmpciatme at which 

1 cntcicd the cippaialus Call thai quantity Q then 

f\ T tT 1"\ 

Q = A ,i (- 1 1 - * h 

vlieic A',, is the mean specific heal ol I he gas between these tern- 
K'latuics, aL the lowei picssuie P,, 

We may define Q as the quantity of heal which each pound of the 
as would liavc to lake up within the appaiatus if iLs Icmpciatuie 
m lca\in Ihc nppaiaLus wcic made equal to its leinpeialuic on 
uliy It measuics Hie available cooling effect due to each pound 
)f gas llial passes thiough the apparatus. 

* Ac Lending to then icsults loi an, Uio fall of tompcratino oxpiossocl m degiecs 
cntigraclo is /">7'i\ 2 

0275(P,- P)(^). 

vhcuo P i and Pjt arc tho piossmes in atmoflphoi cs 



172 THERMODYNAMICS [en 

So long as there is no communication of heat to the gas, by 
theimal mtei change 01 othei wte, while it is passing thiough the 
appaiatus, the gas simply passes off at a lower tempeiatine T'. 
The gas that passes off has the same total heat I as the gas that 
enteis (Art 72), though its tempeiatme has chopped If we were 
to restoie it to the ongmal tempeiature T before letting iL pass 
oft, it would take away moie total heat than it bungs in, the cliffei- 
ence being equal to Q. Its total heat / at exit would then be gi eater 
than its total heat on. admission by the quantity Q, though its 
tempeiatme would be the same. The existence of a Joule-Thomson 
cooling effect in any gas depends on the fact that I he 1otal heat 1 
is a function of the piessuie for a given tempeiatme the total heat 
is gieatei when the piessure is low. 

Suppose now that theie is a counter-cm lent mteiehanger by 
means of which the stieam of gas which has passed the oufice takes 
up heat from the stieam that is on its way to the ounce, with the 
result that the outgoing stieam, before it escapes, has its tem- 
peiatuie lestored to T or veiy near it. Tins is easily accomplished 
by having, within the appaiatus, a long appioach pipe or woim 
through which the compiessed gas passes befoie it i caches the 
ounce, and lound the outside of which the expanded gas passes 
away, i>o that theie is intimate theimal connection bet \\cen the 
two stieams. Foi simplicity we may assume the inteichangei to 
act so peifectl} that when the outgoing gas leaches the exit il has 
acqmied the same tempeiatme 2\ as the entenng gas Each 
Ib of it will theiefoie have taken up a quantity of heat equal to 
Q. as defined above. 

124 First Stage Undei these conditions the appaiatus will 
steadily lose heat at the late of Q units foi eveiy pound of gas that 
passes through If we suppose the appaiatus as a whole to be 
theimally insulated against leakage of heat into it from outside, 
theie will consequently be a continuous i eduction of the stock of 
heat that is held by the pipes and the gas in them The lesult is a 
piogiessive cooling which constitutes the hist stage ol the action 

It may help to make the action cleai ]f we chaw up an account 
of the eneigy icceived and dischaiged bv the apparatus Gas 
enteis at A (fig. 55) undei the piessuie P / and at the tempeiatuie 
T! Gas leaves the appaiatus at B under the piessuie P n and at 
the same tempeiatme T" l5 having taken up, through the aclion of 
the mteichangei, a quantity of heat equal to Q. The pipes and 



iv] THEORY OF REFRIGERATION 173 

expansion orifice arc noL shown in the sketch they aie within the 
enclosing case, which is assumed to be a peifcct non-conductoi of 
heat. Dining the fiist stage of the action the stop-cock C is closed, 
and all Lhc gas that has gone in at A goes out at B, it is onh T \)y 
the cntiy of gas al A and by its escape at B that eneigy enters 
01 leaves the appaiatus 

Each Ib of en Leung gas contains a quantity of internal eneigy 
E i, and the woik that is done upon it as it goes in is P^V A Piach 
Ib of outgoing gas contains a quantity of internal eneigv EU, and 
does woik, against external picssme, equal toP/J 7 /; Hence, for 
each Ib that HOAVS thiough, the net amount of heat which the 
appaiatus loses is 

^ a + P n V a -(E l + P l V. l ), 01 / Z ,-I, 

But the amount so lost is Q, namely the heat that is lequued to 
tcstoic the gas to the lempeiatuie at which it makes its exit. 
Hence Q = In 1 1. 



Fit/ 55 

The contents ol the nppaiatus become coldci and colrlci in "* 
jonscquenec ol this conhmud fibsliaclion ot heat But it is nn- 
soilant to notice that Iheu fall in tempoialiuo does not affect 
he value of Q We assmm thai Hie action ol the lluimal inlci- 
hangci continues to be prilect, in that ease llic exit tcmpeiatuie 
,vill still be e([iial to the initial tcmpcialuu 7\ lio^c\ci cold the 
nteuoi becomes in the ncighbouihood ol the expansions alve. 
riieie will be no change in the \aluc ot either T ]S 01 /,, and conse- 
jucnllv no channc in Q The value of Q, as the abo-sc expicssion 
.hows, depends cntnelv on the conditions at A and at B, with 
KM feet interchange this means that it depends only on P t , PJI, and 
f\ It is independent of any tcmpciature conditions within the 
ippaiatus It is thcicfoic not affected by the piogiessivc cooling, 
ind ictains the same value as the action pioceeds*. 

* It will bo nhown in C'haptoi \ II that tho quantity Q, which moasuics the 
vailablo cooling efloot within the apparatus -when the pressuics P ( and P ]t and 
dmission tcmpfiatuio T\ aro asaigned, can bo calculated jf wo know the coefficient 



174 



THERMODYNAMICS 



[en. 



This staoe of progiessive cooling continues until the temperaluic 
of the gas at the place wheie il is coldest, namely on the low- 
pressuie side of the expansion- valv e, falls not only below the ci itical 
point, but to a value T 2 which is low enough to let the gas begin 
to liquefy undei the pressiue P B . In othei woids T>is the boiling 
point coriespondmg to PB This is the lowest tempeiatiuc that is 
leached 



Fig 56 





Fig 56o Ideal pi ocess ot Regeneiativc Cooling 

A continuous giadient of tempeiature has now become csLab- 
hshed along the flow-pipe within the appaiatus fiom Iho point ol 
enliance, wheie it is 2V to the high-piessuie side of I he cxp.msion- 
\alve, wheie it exceeds T 2 by the amount of the Joule-Thomson 

of expansion of the gas under constant pieasuie foi v.uious piossuios, and also flic 
volume (pei Ib ) for various pictures, at the tompeiatuio T t Wilting V ii i Mm 

volume at any pressure and (^) foi the coefficient of expansion, namely Mm 



iate of change of volume pei unit of change of 
constant, we shall see there (Ait 1S2) that 



mo when tho picssuio IN 






the temperatures being taken as T l thioughout 

Q may also be found e^peiimentaUy, bj observing tho chop ol lompomlmo 
li - 1 which takes place when the gas expands fiom P L to P, t thiough a Joulo- 
ihonison orifice without any interchange of heat 



r] THEORY OF REFRIGERATION 175 

L-op. Theie is also a conbinuoiis gradient along the retain pipe 
onr T 2 , on the low-pressuie side ol' the valve, to 2\ at the exit 
he flow and leturn sti earns aie in close thermal contact, and at 
ich point theic is an excess of temperatme in the flow which 
lows heat to pass by conduction into the leturn, except at the 
itiancc Avheie, under the ideal condition which we have postulated 
' peifcct inteichanqe, the tenipeiatiue of both floAv and retinn 

Z'i 

Tint, state of things is diagianrrnatjcally represented in fig. 56. 
heic the flow and letuin aie icpiescnted as taking place in stiaight 
pes, one inside the othei to piovide loi interchange of heat, 
nlering along the innei pipe A the compiessed gas expands 
nough a constricted onfice E (eqim alent to an expansion-vah e) 
to a vessel fiom which it letinns b}' the outei pipe B The 
5 sscl is piovidcd with a stop-cock C by which that pait of the fluid 
Inch is liquefied can be diawn ofl when the second stage of the 
)eiation has been i cached In the tempeiatuie diagram (fig 50 a) 
r N represents the length of the intcichangci, DM is the initial 
ncl final) tenipeiatiue '1\, GN is T>, and FG is the Joule-Thomson 
op. DF is the giadientfoi the [lew-pipe, and GD foi the letuin. 

125 Second Stage When this giadicnt has become established 
egds begins to hqucf) , the appaiaLus does not become any coldei, 
id the action enters on the second stage, which is one of thcimal 
[iiilibimm A ceiLam small iuiction ot I he gas is continuously 
juc'hcd and may be dunned oil as a liquid Uiiough the stop-cock 

The' laigc'i iiaction, Avhich is nol hquclied, continues to escape 
longh Ihe inlciehangci and to lca\ c (he appaiatus al thr same 
mpcialinc as befoic, namely the tcnipciatine r l\ equal to that 

Ihe enU'iing gas Call this unliqiiefied fi action q, the'n 1 q 
picscnts the fi action that is eliawn oil as a liquid at the tcmpcra- 
ie j 7\ Since lire apparatus is now neithci gaining noi losing 
til on Ihe Avliolc, ils heat-account must balance, horn Avlnch 

IA = ql,> + (1- 0)/ t , 

icrc /j( is the total heat pei Ib of the gas entering at A, IK is the 
tal heat per Ib of Lire gas leaving at B, and /,- is the total heat 
r Ib of the hqurd leaving at C In this steady Avoiking the 
gregate total heat of the fluid passing out is equal to that of the 
id passing in. The fluid, as a whole, takes up no heat in passing 
rough the appaiaLus 



176 THERMODYNAMICS [en. 

Suppose now that the liquid leaving at C were evaporated at 
its boiling point 7\, and then heated at the same piessure fiom 2\ 
to Tj. The heat required to peifoim that opeiation would be 

(1 - q) [L + K, (2\ - TJ\. 

But that hypotlictical opciation would icsult in this, that the 
Avhole oi' the fluid then leaving the apparatus would be icstoicd 
to the tempeiature of entry, namely 2\, sinee the pait which 
escapes at B it> already at that tempeiatme Hence the heat ic- 
quned for it is equal to the quantny Q as defined in Ait 123 We 
theiefoie have 



Q. 

from which l-q= ---, 



This equation allows the fraction that is liquefied to be calculated 
when Q is known' The fiaction so found is the ideal output of 
liquid, for we have assumed that theic is no leakage of heat fiom 
without, and that the action of the inteichangei is peifeet in the 
sense that the outgoing gas is laised by it to the tempciatuic of 
entiy. Under leal conditions theie will be some theimal leakage, 
and the gas will escape at a tempeiatme some\\hat lower than 
T 1 the effect is to diminish the fiaction actually liquefied 

The fiaction 1 q is mcieasecl by using a laiger piessurc-diop. 
It is aKo mcieased by i educing tlie initial tempeiatme T I} thus 
the output of a given appaiatus can be laised by using a sepaiatc 
lehigeiatmg device to pie-cool the gas Pie-cooling is indispensable 
if the method is to be applied to a gas m which, like h> drogen, the 
Joule-Thomson effect is a heating effect at oidmaiy tempciatiucb, 
but becomcb a cooling effect A\hen the initial tempeiatme is sufli- 
cicntty low. 

126 Linde's Apparatus. The punciple of legeneratn c cooling 
descubed m the piccedmg aiticle was fiist succcsslully applied 
by Lmde m 1S95 foi the pioduction of extiemely IOA\ temperatures, 
and for the liquefaction of an, by means of an appaiatus shoA\n 
diagiammatically m fig 57 It consists of an inteichaneer CDE 
foimed of two spiral coils of pipes, one inside the othei, enclosed 
m a theimally insulating case. A compiessmg pump P dchveis 
an under high piessuie thiough the valve H into a coolei J Avheie 

* The specific heat of the vapom is horo treated as constant from T z to T I} 
which is very neaily true at low pressures 



THEORY OF REFRIGERATION 



177 



he heal developed by compiCbsion is lemoved by water circulating 

n the oidinaiy way fiom an inlet at K to an outlet at L The 

uglily compressed air then passes on through the pipe HC to the 

imcr woim and aflcr travel sing the woim it expands through the 

hioltle-valvo ft into the vessel T. Iheieby snffciing a diop in 

cmpciatuic. Then it lelurns thiougli the outer woim F and, being 

n close contact with the innei woim, gives up its cold to the gas 

hat is slill on ils \\ r <iy to expand Finally it icachcs the com- 

ncssion cyhndei P through the suchon-\ alve G, and is compressed 

o go again Ihiough Ihc cycle Dining the first stage it simply goes 

ound and round in this way, but when the second stage is i cached 

iul condensation begins, the pail that is liquefied is diaunoff at V 

nd I he loss is made good by pumping m moic an tluough the stop- 

alvc al A by means of an auxihaiy low-picssurc pump, not shown 




Fig f~>7 Limb's llo^enoialuo Ap|)<uutii^ 

L the sketch, which dchveis an lioiu the almospheic to the low- 
lessiue side ol the ciiculatiiig system 

Liude sho\\ed I ha I bv keeping this lowei picssuic fanly high, il 
[)iactieable to icducc the amount of uoik thai has to be spent 
hquel\ing a gi\en quantity ol air He pointed out that while 
ic cooling clled of expansion depends upon the diffeicnce of 
essuus PI and Pn on the two sides of the expansion-valve, the 
oik done in compressing the an in the ciiculatmg system depends 
i the ratio of /' / to P/ ; . Il is loughly piopoi lional to the logarithm 
' I hat witio, (01 il approximates to the woi k spent in the isothermal 
mipicssion of a pcifecl gas, which (by Art 28) is 7?Tlog t ?, 
here ; is the lalio of the volumes or of Lhc piessnics. If, for 
:ample, P., ib 200 atmospheics and Pn is one atmosphere, the 
lolmg ell'cct is proportional to ]99 and the work of the compiessmg 
imp is 10 uglily proportional to log 200. If on the othci hand the 



13 T 



12 



178 THERMODYNAMICS |c-ir 

back piessiue Pn is 50 atmohphcics Ihc cooling elTicl is piupor- 
tional to 150 and the woik of the main compicsMng-pump (o log I 
The cooling effect is icduced by only abonL onc-lbmlh, while Ihc 
work is leduced by nearly three-fourths ALLci allowing I'm Uu i 
extra amount of work that has, in the second case, lo be s|>enl on 
the auxiliai}^ pump in supplying an at 50 nlniospheies lo replace 
the fraction which is liquefied, there is still a marked adxanlage, 
in point of theimodynamic efficiency, m using a closed cycle \\ilh 
a modeiately high back piessure. 

The Lmde process is employed on a commeicial scale lo liquefy 
air as a first step in the separation of its constituents A Linde 
plant at Odda, m Norway, liquefies about one hundied tons ol 
air daily for the pmpose of supplying nitiogen I'oi use- in I he inanu- 
factuie of cyanamide, an arhlkial nitrogenous t'erlili/ei \\Iueli is 
formed by passing gaseous nitiogen over hot calcium cui bide The 
method by which the constituent gases aie separated will be 
presently desciibed. 

127 Liquefaction of Air by Expansion in which Work is 
done. Claude's Apparatus. The chop in lunpualme which a 
gas undeigoes in parsing fiom a legion of high piessmv lo ,1 i< uion 
of low piessuie would be gieatcr if Ihc piociss \u-i< eniiduclcd 
ie\eisibly, as b> expansion m a cyhndci in whuli llu- "as ( | l)OS 
mechanical woik We should still luu c Ihe small Joule-Thomson 
cooling effect, but in addition thcic would be Ihc (n t nei.illy much 
laigei) cooling effect that is clue to the eneigv \\lueh (lu i>as loses 
in doing woik Eail> attempts made by Siemens, Sol\ ay, ami olhc is 
toieacrMery low tempeiahnct, by applying a lheiiu.il iiileich.innci 
to an expansion cylinder, failed mainly because the e \hude, S.MMI 
leached a tempciatuic at which the lubiicaul ho/e Tins dilliriilh 
^as successfully oveicomc in 1902 by Clamh , u ho (oiind Ih.il I he 
difficulties attendant on expansion m a woilunn cvluulf i ,|, mn to 
a tempeiaiuie below the cnt.cal point of an could be ove u-omc by 
using certain hydiocaibons as lubucants A I M (Imc.iilxui s,,( h ,s 
petioleum-ethei does not solidity but rcm.uns N ,scous nt u-iu- 
peratuie as low as - 160 C Usm a a lubueant of llus lund (l, 1K Ie 
succeeded, as an expcumcntal low dc (<ncc, ,., ], ( ,,,ef\'m.. an- , 
an expansion cylmder furnished wilh a icgemn.hve rmmh,- 
cuiient thermal mteichangei the cxpaus.on cyluule, snnnly 

taking the place of the expansion-vnl V eMn ana 1 , 1 )arnlussuchas Ihal 
of Art. 120. He also found that Ihc liquid, once it h, ms to fo, n 



IV] 



THEORY OF REFRIGERATION 



179 



serves itself ns a lubricant, and no other need then be supplied 
Under these conditions, however, there is little if any advantage 
in using an expansion cyhndei, for the volume of the fluid at the 
lowest exticme of tempeiatuie is so small as Lo make the woik of 
expansion insignificant. Theie is not much additional cooling, at 
the same lime it is far less practicable to secure theimal insulation 
with an expansion cyhndei than with a Joule-Thomson ounce. 
Claude subsequently obtained a more economical icsult by giving 
the appaiatus the modified foim shown in fig. 58 In that airange- 
nicnt pait of the compressed air expands in a woikmg cylinder to 
a tempeiatuie which may be just below the critical temperatuie, 
and the an which is cooled (but not liquefied) by that expansion 
is used as a cooling agent on the remaindci of the an, with the 




csull lh.it soim- o( the lalUi is liquefied nuclei the higher pussiuc 
il \\luch it is supplied The stipph comes in, al a picssiue of 40 
ilmosphcics 01 so, lluough the eenlial pipe of Ihe coimlei-cmient 
nlc'icluingci M Pait of it passes mlo the expansion C3 r lmdcr D 
vlit'ie it expands doing work, and is then chschaigcd thiough the 
iondcnsmg \ essel L, wlieieil sei \ es as the cooling agenl lo maintain 
i lempeialiuc .somewhal Ie)wei than H0 C., the cntical tcm- 
K inline ol an The lemaindei of the eompicssed an enters the 
nbcs ol L and is condensed thcic, undci piessme, chopping as a 
iqmd mlo Ihe chambei bclo\\ fiom which it can be chawn off 

In ji I'nilhcr development of Uns invention Claude made the 
'xpansion compound, and caused the expanded gas to act as a 
ooling agent after each stage, becoming itself \varmed up in the 
>roccss The expanded gas is thereby picpaied to suffer further 

122 



ISO 



THERMODYNAMICS 



[en 



expansion without an excessive fall of tcmpeiatuic. Dmmg Us 
expansion the gas in the C3 r linder is not so neai the liquid state 
as to make expansion in a woikmg cylinder of hi tie use The 
arrangement with compound expansion is illustiated in fig 59. An 
undei picssuie enters, as before, tluough the central pipe of M. 
Part of it goes to the fnst expansion cylindci A, docs ivoik there, and 
piocceds at i educed prcssme, and ata tempeiatuie below thccnlical 
point, thiough the outer vessel of the condenser L lt in themnci tube 
ol which some of the compicssed an is being condensed This waims 
up the expanded an to some extent, and it then passes on to 
complete its expansion m B, which again brings its tcmpcialine 
down sufficiently to allow it to act as condensing agent for the 




Fig 59 Claude's Intel method Aulh compound expansion 

remaining poition of Ihc an under piessine, in I he second con- 
dense! L 2 This division of Ihe expansion into I wo (01 il nuiy be 
moie than two) stages is equivalent to making Ihe piocrss as a 
whole more nearly isolhcimal, so IhaL Ihe an need not al any st.igc 
deviate veiy widely fiom a tcmpeiahuc which is just sulhcicnllv 
below the critical point to allow liquefaction to go on undei Ihc 
pressure at which the an is .supplied 1 ' 

128. Separation of the Constituents of Air. The lique- 
faction of an enables the constituent gases to be sepaiatcd because 

* G Claude, Gomples Rcnrlus, II June !')()(>. and 22 Of, 1000 Hen H!HO Jus book, 
on Liquid An, Tis f'oUioll, 191 { An iuhclo by PJLOOHSOI E Matluas in RcnLc. 
g&t&ale dcs Si icnccs, 15 SopL 1907, contains un mtcioatmg aocounb of ilu> wliolo 
subject of ilie indusbnal liquefaction ot au 



v] THEORY OF REFRIGERATION 

n ic evapoiation they have diffeient boiling points The boiling 
DOint of mtiogen, under atmospheric pressure, is about 195 C 
n 18 lower than that ot oxygen, which is - 182 C. When a quan- 
,ity of liquefied an eAapoiates fieely both gases pass off, but not 
n the original piopoihon in which they aie mixed in the liquid 
The mtiogcn evapoiates more leachly, and the liquid that is left 
jecom.es iichei in oxygen as the evapoiation proceeds This 
lifference in -\olatihty between oxygen and nitiogen makes it 
possible to cany out a piocess of rectification analogous to the 
jiocess vi Inch is used by cbstillcis foi extiacting spuit ii^m. the 
'wash" 01 lei merited wort, which is a weak mixtuie ot a nhol 
ind \vatci, by means of a device known as the Coffey Still 

In the still patented by Aeneas Coffey in 1830 theie is a rectifying 
jolumu consisting of a tall chambei containing many zig-zag 
shelves 01 bafllc pLites The wash enteis at the top of the column 
ind tuckles slowl} down, meeting a cuiient of stearn which is 
idnuttcd at the bottom and use-, up thiough the shelves. The 
lown-commg wash and the up-going steam aie theieln brought 
into close contact and an exchange of fluid takes place. At each 
-.tagc some of the alcohol is e\apoiated fiom the \\ash and some of 
the steam is condensed, the heat supplied b\ the condensation of 
UK steam seivmg to e\apoiate the alcohol The condensed steam 
becomes pa.it of Uie do\\n-coming sticam of liquid the e\ apoiated 
ilcohol becomes pait of the np-gomg sticam of \apom Finally 
it llic top a Acipour compaialix eh nch in .ilcohol passes off at 
the l)oltom a liquid accumulates \\liich is watci \\ith little 01 no 
ilcohol in it A tcmpualuic giadicnt is established in the column 
it the bottom Ihc Umpeiafuic is llmt ol steam, and at the top 
Ihc'ir is a lowei knipcialiuc appioxunating to the boiling point 
it alcohol The wash enti-is at this compaiatively low tempeiatuie, 
,uul lakes up heat horn Iht steam as it tnckles down 

Lnidc applied the same geneial idea in a device foi bepaiating 
Lhc less volatile o\\gen horn the moie volatile nitiogen of liquid 
in In this de\ice, the pnmaiy puipose of A\ Inch \\a^ to obtain 
Dxygen, theie is a iccti lying column down vihich liquid an tnckles, 
-,Laihng at the Lop at a tempeiatuie a little undei - 19J. C 01 
70 absolute, \\luch is the boiling point of liquid an nuclei at- 
mosphei ic picssmc As the liquid tiicklcs down it meets an up-going 
aream of gas which consists (at the bottom) of neaily puie oxygen, 
initially at a tcmperatuie of about 91 absolute, that being the 
boiling point of oxygen undei atmospheiic pressure As the gas 



182 



THERMODYNAMICS 




A 



A 



uses and comes mto close contact wilh the clown-coming 
liquid, theie is a give and take of substance at each shii>(' 
some of the using oxygen is condensed and some ol Ihc 
in the down-coming liquid is 
evaporated, the liquid also be- 
comes lather wannei BY the 
time it i caches the bottom iL 
consists of neaily pure oxygen 
the nitrogen has almost com p 1 et cly 
passed off as gas, and the gas 
wlir i passes off at the top uon- 
SIL jveiy largely of mtiogen Moie 
piecisely it consists of mtiogen 
mixed with about 7 pei cent of 
oxygen in other woicls, out of the 
whole oiigmal oxygen content of 
the au (say 21 pei cent ) two- 
thuds aie biought do\\n as liquid 
oxygen to the bottom of the 
column, while one-thud passes 
off unsepaiated along with all 
the mtiogen The oMgin that 
gatheis at the bottom is with- 
drawn foi use, and is c\ apoiatcd 
in sen ing to hquefv fiesh com- 
pressed an, which is pumped into 
the apparatus to undergo the pi o- 
cess of separation The cold 
gases that aie leaving the ap- 
paratus, namely the oxygen 
which is the useful pioduct, and 
the mtiogen which passes off as 
wastegasatthe top of thecoluirm, 
aie made to ti averse counLer- 
cunent mtei changers ou then 
way out, so as to give up then cold 



B 




F 



C) 





'////////////777T?//'/'//', 



l fl)I 



to the incoming compressed mi ]PIR ()0 LimItfllll|lpllMilllH , 
that is on its way to be liquefied o\inuLin{- o\y^n I>y icdiduiiiou 

In the chagiam, fig GO, these conn I ci -cm ion I mloic lumens ,11 1 
omitted for the sake of clearness, bul I he csst-nlial Icaluus ol llu 1 
condensing and lectifymg apparatus are shown The (igiue is 



v] THEORY OF REFRIGERATION 183 

Dased on one given m Lmde's patent of June 1902, which descnbes 
he m\cntion by which a process* of lecLiiicalion has been success- 
'ally applied in the ex ti action of oxygen fiom an. 

The ic A is the rectifying column, consisting m tins instance of 

i vcitical chamber slacked with glass balls, through the mteistices 

)f winch the liquid tuckles clown The lowei pait B contains an 

iccnmulation of fluid which, when the appaiatus has been at work 

ong enough to establish a unifoim legwie, consists of neailv pine 

iqnid oxygen Compressed an, which has been cooled by passing 

Jnough a co untci-c uncut mtcich.ingcr, entcis at C, becomes 

iqucficd in the veitical condenser pipes D, which aic closed at 

he top, and diops dovn into the vessel E It gnes up its latent 

icat to the oxygen in B, thcieby evapoiatmg a pait of that, and 

,o supplying a sticam ol gaseous oxygen which begins to pass up 

he icclif>mg column. On its wa\ up, this stieam of gas effects 

in exchange of maUual with the liquid an which is tiicklmg 

lo\\n gasious oMgen is condensed and leluins with Ihe sticam 

o the \ esse I B, while mtiogen is (.\apoialcd and passes oil at the 

op ol I he column, al A r , mixed \\ilh some o\\gen The escaping 

jas goes lhioiiL>li an inlciclumgci, taking up IK at liom the m- 

ommg compiessed an 

The a cci mml.il 101 ioL neaiK pine liquid oxNgcn m//o\ cillows into 
lu lo\\ci vessel F, wh< ic a suppkniuilai \ supply of compiesscd an 
nUimg al (1 is employed to <_\apoiale it In me, ins ol a siimlai 
u languni ill ol condense i lubes open al the bollom and closed 
il UK lo]), this .in bccominii ilsell condensed in I he pioci'ss, anil 
allmg as a liquid into Ihc \esstl // The limited an fioni I* 
ind tiom // is slill undiM picssiiit il passes up tliiouyli t \pansion- 
, ,il\ es K lo I lie lop ol Ihc icclilving column, win le il is dischaiged 
>\(i UR ylass balls al ii pussiiu nol makiialh abo\L thai c>( the 
ilmosphuc This secmcs llu incessai} dilftuncc in k-nipiia- 
inc bclucen Ihe bottom and top of the column The com- 
>tisscd an l)la^s Ihe pail ol hc.itci and i\ apoialoi ol Ihe liquid 
)x\gen <if the bollom. at the compaialivcly high lempeiatme of 
iboul 01 absolute, be loie ituiicleri^ocs uctification Inolhei woids, 
L nol only coiicsponds to Ihe "wash" of Ihe Cofley still, but it 
ilso sei\es as the equivalent of Ihc hcatci by which the liquid at 
[he bollom of the still gi\Cb off an upwaid cuiuat of steam 
[Jascous oxygen, Ihe dcsiied pioducL in this case, passes off al 0, 
ind like the waste gas, consisting mainly of niliogen, which escapes 
ifA T , it goes tluough a counter-cuuent mterchangei, taking up heal 



184 THERMODYNAMICS [en 

fiom the compiessed an which enteis partly at Cand paitly at G 
It is the waste gas in this pioeess that foum Lhc analogue of the 
lectified spuit Avluch is the useful pioduct of the Coffcy still. 

At first, when the machine begins woikmg, Lhe air is highly com- 
piessed, but aftci the opciation has gone on foi some Lime, and a 
steady state is appioachcd. a much lower piessme is sufficient IL 
must be high enough to make the an liquefy at the IcmpciaLuie of 
the liquid oxygen bath, say 91 absolute, ami in pincticc il is kcpL 



100 




10 20 30 40 50 60 70 80 

PERCENTAGE OF OXYGEN IN LlOLI/D 

F,g (>1 

higher than this to emiuc that Ihe diop in lcni|K ruLmc al Liu 
expansion -vahe may be sufficient to make good any losses due 
to leakage of heat fiom outside, and to impelled mlci change in 
the counter-curient appaiatus Foi sonic time, after I he iippaialns 
is first staited the icctifying aclion is impel (Vet, bul as I lie pioocss 
goes on the liquid contents ot the vessel Ji become i ichei and 
nchei m oxygen, the icctification becomes moie complete, and Ihe 
piessure maj be icduced Unclci piaelical condilions it is easy lo 
secuie that the gaseous pioduct shall lie pine lo iho extent of 
containing 98 per cent, of oxygen 



] THEORY OF REFRIGERATION 185 

129 Baly's Curves The action of the lectifymg column will 
* ma.de moic intelligible if we icfer to Lhe icsults oi cxpeiiments 
ibhshed in 1900 by Baly*, which deal with the nature of the 
apoiafion m mixtuies of liquid oxygen and mtiogen. Given a 
ixtiue of these liquids in any assigned piopoition, equihbiiimi 
tween liquid and vapoui is possible only when the vapoui 
ntains a definite piopoition of Lhc two constituents, but this 
oportion is not the same as that in the liquid mixtme Say for 
ample that tJic liquid mixtuic is half oxygen and half mtiogen, 
en according to Bah 's cxpeiiments the vapoui pioccedmg from 
ch a mix lure will consist of about 22 per cent of oxygen and 
i pci cen I. of mtiogen With these pioportions theie will be 
[iiilibiium If howcvei a vapoui iichci than this m oxygen be 
ought into contact Avith the half-and-half liquid, pait of the 
seous oxvgen will condense and pait of tlie liquid mtiogen 
ill be cvapoiatccl, until the piopoition giving equilibiium is 
aclied The cm \ e, fig. (il , shows, toi each piopoition in the mixed 
1 1 ud, \\hat is the coiicspondmg piopoition m I lie \ apom nccessaiy 
i 1 equilibimni in olhei woids wlwt is the pioj)oibou which the 
nslitucnls luu c in the v.ipoui, when that is beiny J'oimed b"s 
apoi ilion of the mixed liquid, in the h'isl slayes ol such tin c\ti- 
>iat ion, befoie UK piopoi lion m the liquid changes In I his cin\ e 
e base-line specifics I he piopoition ol ox \ gen in the liquid nn\ tin c, 
)iu lo 100 pel eenl , <ind the oidmales <>i\ c the piopoihon of 
\gcn in I he coiicspondmg \apom, A\hcn I he \fipom is (oinud 
ide i .1 piesMiic equal lo lluil ol Ihe almosphcic Mueh I he same, 
ncial ielalu/11 \\ill hold al olhci piessuies 11 will he seen fiom 
e eiii'\e' lh.il when Ihe e \ apoialing iKjiud mi\lme is liquid an 
\vgcn 2^ pel cenl , mliogen 79 pel eenl ), the piopoilion ol 
yyen piest nl in Ihe \ apom I hat is ( oming off is aboul ? pei cent. 

a hi lie less. 

This is what oeeius al Ihe lop of the lecliiVmg column in the 
)j).uatus ol lig (!() The Juniid lhat is e\ apoitiling Iheie is lieshly 
imed liquid an, and he net Ihe waste gases cany ofl about 7 pei 
nl of oxygen Coming down Ihe column Ihe liquid hnels itsell 

eonlael wilh gas containing moic oxygen than concsponds to 
uihbiumi Acfioidmgly oxygen is condensed and mtiogen is 
apoiatecl al each stage in Ihe descenl, in Ihe elloil at each level 
a condition ol eqmhbiium between the liquid and the 
wilh wJuch it is there in contact. 



Ba\y,Plul May ,\o\ xux, p 017, 



136 



THERMODYNAMICS 



[cir 



Fig 62 is another foim of Baly's cuive, the form., namely, 111 
which the icsults of the experiments wcie ongmally shown. Thcic 
the oidinates icpicsent the absolute tempcialmc (m centigrade 




30 4O r .O nO 70 

PERCENTAGE OF OXYGEN 



Fig 02 



degiees) at \\hich, under almospheiic prcssinc, llu- i in. \c-cl li(|iii(l 
boils, and two ciuvcs aic dia\\u which show by IIKMM.S ol I he scale 
on the base-line the percentage consLiLulion ol (1) Ihc li(|iml, 
(2) the vapom, when the condition ol. cquilibnuni bcLvvtcn lupud 



THEORY OF REFRIGERATION i 8 7 

L vapoiu is attained*. A honzontal line drawn across, the curves 
any assigned level of tempeiatiue shows the composition of 
)our and liquid icspeclively foi that tempeiatme, when the two 
in cqnilibiium Taking an inteimediate point between the top 
I botlom of the lectil'ymg column, and di awing the hue for 

coiicsponding tcmpeiatuie, we should find the lespective 
vposi lions of liquid and ^ipom theie to appioximate to the 
ucs found fiom Ihe two cunes, this appioximation being closer 

moie slowly tlie liquid trickles clown, and the moie intimate 

contact between liquid and gas 

f a simiLii condition of equilibrium holds at each stage in the 
cess of liqnclyjng a nuxtuie of the gases, these cnives may also 
taken as showing what is the piopoition of the constituents in 

mixed liquid at each stage \AhiIe condensation ol the mixed gas 
icccds Thus when an containing 21 pei cent of oxygen begins 
liquefy, the liquid initially loimecl should, nuclei equilibimm 
idihons, be much nchci in oxjgen the piopoition ol ox\gen 
il, accojtlmf* to the cui\e is 18 pei cent. 

These conditions aie appioximateh icahzed when the piocess 
)\vn as "suiibbmg" is icsoited to m the liquefaction of an. 

I his piocess, which will be presently descnbed in the toim in 
uh it Jias been piacticalK tamed out by Claude, a paitial 
MKihon bclwccn the two constituents is clfected dining the act 
liqiK. I'.iclion. 



/ing li^me^ .ue ffivou 


by Baly 




Absolute 


Poicintagc of 


OM o;cn 


TompoiahiK 1 


In Vapoui 


In Liquid 


77 5 t 








78 


2 18 


8 10 


79 


80 


21 GO 


80 


1200 


33 35 


81 


1760 


43 38 


82 


23 60 


52 17 


83 


2995 


5955 


8t 


3686 


6G20 


85 


4425 


7227 


86 


52 19 


7780 


87 


0053 


8295 


88 


09 58 


87 GO 


80 


7945 


91 98 


90 


8980 


96 15 


90 96 


100 


100 



188 THERMODYNAMICS [en. 

130 Complete Rectification. In Linde's invention of 1902 
the rectifying pjocess is incomplete, for alt ho ugh. the piocess yields 
neaily pure oxygen it leaves a pait of the oxygen to escape 
m the waste gas and it does not yield pine mtiogen. In a commer- 
cial piocess foi the manufactuic of oxygen this is of no consequence 
foi the ia\v matenal costs nothing, and theniLiogen is not wanted. 
But a modification of the piocess enables the sepaiation Lo be 
made substantially complete, should it be desiied to complete it, 
and allows appioximately pmc mtiogen to be obtained, as well as 
pure 0x3 gen. 

The modification consists in extending I he rectifying column 
upwaids and in supph ing it at the top with a liquid nch in mtiogen. 
A fiactional method of liquefaction is adopted, which scpaiates 
the condensed matenal at once into two liquids, one containing 
much ox^ygen and the othei little except mtiogen The lattei is 
sent to the top of the icctifying column, while the foimci enleis 
the column al a lowci point, appiopiiate to the piopoilion it 
contains of the two constituents PiacLically pine mtiogen passes 
olf as gas at the top, andpiactically pine cm gen fiom the bottom 

Fig 03 is a diagiam showing this modified piocess in a lonn gi\ en 
to it by Claude. The countei-cuiient mtcichangcis which aie ol 
couise pait of the actual appaiatu-, aic omitted fiom the dicigiiim 

Compicssed an, cooled bv the mteichangei on its way, uileis llu 
condensci at A. The condensci consists of two sets ol \cilicril 
pipes, communicating at the top, whcie they all open into the vessel 
B, but sepaiated at the bottom. The central pipes, which open 
fiom the vessel A, aie one set the othei set foini a ling lound I hem 
and chain into the vessel C Both sets aic nnmeised in a bdlh, .V, 
of liquid which, when the machine is in full opeial ion, consists ol 
neaily pine oxygen The condensation of the compicsscd an causes 
this oxygen to be evapoiatcd Part of it slicanis up ihe icchlymg 
column D, to be condensed thcic in caiivmg out the woik ot 
lectincation and consequently to iclum to the vessel below The 
lest of the evapoiated oxygen, loimmg one of the useful pioduets, 
goes ofl by the pipe E at I he side In these fcatmcs the appaialus 
is substantially the same as Linde's, but Iheie is a diflcrenec in 
I lie mode of condensation of the compressed an Euleimg at J it 
fiist passes up the ccntial gionp ol condenser pipes, and (he liquid 
which is fonned in them contains a i datively huge piopoition ol 
oxygen. This liquid chains back into the vessel A, when it collects, 
and the gas which has smvivcd condensation in these pipes goes 



THEORY OF REFRIGERATION 



189 



110 



itf^-L^! 

^u,^ 

^_____^, 



n,^ 

HjLMMTg^g^TBB-iJ Ira 



D 






N 



CHI 




COMPRESSED 
AIR 



J'i Claude's apparatus foi the complete sepaiation oi oxygen and mtiogcn 



190 THERMODYNAMICS [e-n.iv 

on thiough B to the outei set of pipes, is condensed in I IK-MI, and 
drams into the othei collecting vessel C It consists ahuosl wholly 
of mtiogen Then the liquid contents of C aie taken (1 1 iron nil an 
expansion-valve) to the top of the lectifying column, win If those of 
A entei the column lower clown, at a level L, chosen lo com spoiid 
with the proportion of the constituents The result is lo secmc 
piactically complete lectification, and the second proeliu'l of Ihe 
machine commeicially pure mtiogen passes off a I Hie lop 
thiough the pipe N and may be collected foi use 

The action m the central pipes ol the condenser i.s lo be mlei- 
pieted in the light of Baly's cuives. The first portions of Hie au- 
to be condensed tuckle down the sides ol Ihesc pipe's and me 
" scrubbed " by the an as it ascends that is to say they aic bioughl 
into such intimate contact with the ascending an that ;i condition 
of equilibrium between hquiel and vapour is al least closely ap- 
pioximated to. The condition of cqinhbiium when gases of I he 
composition of an aic being condensed icquiies, as we. have 1 s( t n, 
that about 18 pei cent, of the hquiel should consist of oxygen 1 . 
Accordingly the liquid which collects in the vessel A is ol I his 
degiee of iichness, 01 neai it And by making the conelensei pipes 
long enough it is cleai that little 01 no oxvgen will be lc-1 1 lo pass 
over thiough B into the othei pipes It is Irue of ceniisc Ilia! in 
the uppei paits of the cential pipers the liquid lhal is (01 ined con- 
sists largely of mtiogen, but as this trickles chw n the pipe- in \\ IIK h 
it ha^ been condensed theie is a give and lake be I ween if and I IK 
ascending gas, pieeisely analogous lo Hint which oceuis in .1 i hl\- 
mg column and when the liquid i caches Ihe bollom il h.is b< c n s> 
much emiched in oxygen as to be neatly 01 completely in <qmli- 
bnum \\ith the gaseous an, and theielore ce>nl.uns about IS p< i e< nl. 
When the 18 pei cent liquid fiom A is disciplined lluoiinh 
an expansion-^ Ive into the lectilying column al L, il piodm es ,-in 
atmospheie which has the composition ol air (21 per cuil ol 
oxygen) Hence the part of the column which e\kncls ,iho\< I Ins 
point has foi its function to leducc the percentage ol ox^nen m 
the ascending gas fiom 21 pei cent lo ml, and lln^ is dom^n Ihe 
second stage of lectification, by means ol the liquid f umi (' \\lneh 
consists almost wholly of mtiogen |. 

* That piopoition, as has been pointed out, in spouting of 15,ily'n cm v<--> M lnf< n 
to experiments made at atmobphouc piessma At Iho hi<.lu,7 pn HSIIKI inuli'i winch 
condensation takes place m Claude's appuiatus it may not bo o\nc Uy Uin siinic 

t Forfuithei paiticulais of some of the subjects tmitod jn Ihm (.'hiiplm icloicmo 
should be made to the authoi'a book on The Mcdiamud Pwdmlwn oj ( '<,!,! 



CHAPTER V 



JETS AND TURBINES 



31. Theory of Jets. We have now to coubidei the manner 
Inch u jcl is foimed m Lhc dischaigc, Ihiough an ounce, of steam 
1113^ olhei gas undei piessiuc To simplify mattcis it will be 
mied Lluit Lhc lluicl Lakes in no heat and gi\ CD out no heat to 
LI bodies dining the opeialion, in othei Avoids thaL the jet is 
nod nuclei adiatheunal eondiLions Suppose a gas to be flowing 
-Hiyh a nozzle 01 channel of any ibim, 1'ioni a icgion \\hcie the 
-.sine ib iclatis el}' high Lo one where it is lowci Each element 
he slieam expands, and Llie woik ^hich il does in expanding 
s cncigy of nioLion Lo the 
iient in fionL of it The \\holc 
ain Lhoieloic acquni's \doeit\ 
he piocess and also incieascs 
olume Lit A and /? (hg (Jl) 
nuayinaiy pailitions, aeioss 
c'h il llous, lalv( u a I ugliL angles 
lu- clneclion of the slieani lines, 
ic inn in I he ngion of highei 
sine Lei /* he tlie piessinc 

'/, v a Ihe \elocil \ Iheu, and V a Ihe volume 
uLily ol Ihe gas has as it passes I he imagmaiv paililion aL / 
il.ulv let I'f,, v b and }\ be the picssnu , velocil\, and \ohimcof 

(pi.mlity at IL Let E a and E b be the mlcinil enei^\ oJ' the 
<U ^L and B iespeeli\ r t 1> In llo\\mg fiom A lo Ji the \elocily 

igcs iiom v a lo v t , and Lheie is eonsequenlly a gain ol kmelie 

gy amounting, j)cr unit of mass, Lo '' t -'- . 

^to 

aeh unit quanlily of gas that enters the space between A and 
is wnk done upon it by the gas behind, amounting Lo PJ r a 
lapsing out of this space at U it docts work on the gas in liont 
lulling to PjfVj, In flowing iiom A to B it loses internal 
gy amounting to E a E b Hence by the pimciplc of the 




Fin (,(- 



nniL 



192 THERMODYNAMICS fen 

conseivation of energy, since by assumption no heat is taken in 01 
given out, . 2 _ 7 , 2 

* - ^ = E a ~E b + P a V w - P b V b (1). 

But E a + PJ\ is /, tlic total heal at A, and & + PJ'\ is / 6 , 
the tobal heat at 7?, and the equation ma\ consequcnlly be wiittcn 

u lT ~~ U a" _ r r / \ 

-- ' a -'a" 'ft H 

"""to 

The gain in kinetic encigj is theicfoic equal to the Joss ol' lolal 
heat, or what is commonly called the l heal -drop." We aie 
ti eating E and / as if they weie expiesscd in \voik units when ex- 
pressed in heat units they have to be multiplied by the mechanical 
equivalent J 

The equation applies as between any two places in the How, 
and taking the piocess as a whole, fiom the initial condition in 
which the velocitv is Uj and total heat 1^ to the final condition in 
which the velocity is v z and total heat / 2 we ha\ c 
-, a _ ~, 2 



2g 

In mairy piaetical cases the initial velocity is zcio 01 negligibly 
small, and then W 2 

^-'x-', , 

wheie v is the velocity acquiicd in consequence of the heal -chop 
This is the fundamental equation fiom which lo calculate the 
velocity which an expanding fluid acquncs in a jtl, slailing fiom 
lest. 

So far theie has been 110 assumption as lo absence of losses 
tin on gh faction 01 eddy cm rents II we assume, as an ideal ease, 
that in the foimation ol the jet the lluid is expanding iiudti such 
conditions that thcie is no conduction of heal lo 01 horn 01 \villun 
the lluid and also no dissipation ot energy lluoiigh fuel ion 01 
eddies, the heat-diop in the equal ion 

o o 

V 2 "-Vf_ 

JL i JL i> 

2fl 

O 

is thaX which oceiirs in expansion with constanl cnliopy We have 
alreadyl seen (Art 80) that this heat-drop is equal lo Ihe area 
\of the ideal indicator diagram (fig G5) lor aehabalie ex- 



pan&ion %om the initial to the final slate, 01 VdP. 

. (5). 



JETS AND TURBINES ]93 

This icsiilt might also be infened fiom the fact that, under the 

sumed conditions, the gas is 

ang all the work of which it is 

Gaily capable, as it expands fiom 

efhst to the second state, in giving 

nctic energy to its own si i earn 

legam of kinetic eneigy is, thcie- 

LC, equal to the aiea of the ideal 

hcator diagiam. 

Assume that we may, with sufficient accuracy, expiess the ex- 

nsion in the ideal indicator diagiam by a foimuLi of the type 

7m constant. Then the area of the diagiam, namely 

l VdP - (P V P V \ 
- ( 1/1 ~"^ K2) 




m 1 

'iicc when the expanding fluid staits fiom icst, at piessme P, 
foim a jet, we have 



v* _ 
"2g m 



r = 



m [ /Pr" 1 



Zl l-^J-JP^, 



an c'(]iialion fiom which to hnd the \elocity t' when NIC 
s f.illen lo iinv lowci piessiuc P, undci the assumed coiidiln>us of 
u \vithoul fnction 01 eddies and with no conduction ol hcdl 
[iialion (f5) is i paiLicuLii case of Equation (1), namely the case 
icic the expansion is isentiopic and wheie the icLition of piessine 
volume in isentiopic expansion admits of being cxpicssed by 
3 formula PV m = constant 

[32 Form of the Jet (De Laval's Nozzle) As expansion of 
i fluid in a jet piocecds the volume and \ elocity both mciease 
is easy in Iriclionlcss adiabatic flow to calculate both, and in 
it way to deleimme the proper foim to give to the nozzle 01 
iiinel, to make provision foi the increased volume, having icgaid 
the mcieased velocity. At any stage the aiea of cross-section 
the channel lequned for each Ib of fluid dischaiged is equal to 
i volume pei Ib divided b) r the velocity. It is convenient to 
kon the area of section per unit of mass m the dischaige, and 
er wards multiply by the numbei of Ibs or kilogrammes. 
ra T 13 



194 



THERMODYNAMICS [en. 

Let M repiesent the dischaige, namely the mass which passes 
throuoh the nozzle pei second, X the aiea of cross-section of the 
stream at any part of the nozzle, v the velocity theie, and F the 
\olume of the fluid there (per unit of mass), then 
,. vX , X V 



On making the calculation foi a gaseous fluid staitmg iiom icst 
and chschaiged into a icgion of much lower pressuie, it will be 
found that m the eailiest stages the gam of velocily is iclativcly 
gieat, but as expansion pioceeds the mcicasc of volume outstnps 
the mciease of velocity The lesult is that the laLio of volume to 
velocity at fiist diminishes, pass.cs a minimum value, and then 
inci eases, and hence the channel to be pioMcled ibi Ihc dischaige, 
aftei passing a minimum of cross-section, expands m the later 
stages. The piopei foira for the nozzle, to allow the heat-diop 
con expending to a laige diop in piessuie to be utih/ed cis fully as 
possible in gnmg kinetic eneigy to the stieam, is Iheiefoie one m 
vhich the area of section at fiist contiacts to a nairow neck 01 
''thioat" and aften\aids becomes cnLugcd to an ex I cut lhat is 
detei mined by the available fall of piessuie 

It is on this pnnciple that De Laval's "convci<><.ut-di\ agent' 1 
nozzle (fig 66) is designed The thioat, 01 smallest section, is ap- 
proached thiough a moie 01 
less lounded entiance which 
allo\\s the stieam lines to con- 
veige, and fiom the thioat out- 
waids to the dischaige end the 
nozzle expands m any giadual ln k' Wl 

mannei, geneially m fact as a simple cone, unlil .in aic-a ol su-hon 
is leached which will coiiespond to the piopei aicn ol discliaigc- 
foi the final -\olume and \elonty, the values of which depend 
upon the final piessuie 

The divei gent tapei fiom the throat oinvaids is made sullieienlly 
giadual to preseive stieam-line motion as completely <is is piaclic- 
able, and so avoid the foimation ol eddies which would dissipale 
the kinetic eneigy of the stieam A vciy shoil lounded enhance 
to the thioat is sufficient to guaid against eddies in Ihc eonvc igent 
portion of the stieam, but m the divergent portion a much moic 
giadual change of section is icquired. The nozzle shown m the 
figuie was designed foi an initial piessme of 250 pounds per 
sq inch and a back piessme of about 1] pounds. liy the back 




r] JETS AND TURBINES 195 

wcssuie is meant the pressure in the space into which the fluid 
s dischaiged. 

In the design of such a nozzle the purpose is (1) to make the 
hschaigc have a given value, and (2) to give the stream as hjgh 
i final velocity as possible by utilizing completely the energy of 
he fluid m expanding down to the back pressuie The data for the 
lesign are the initial pressuie, the back picssure, and the intended 
nnount of the dischaige It Mill be shown as we piocecd that the 
irea of section at the throat depends only on the initial pressuie 
aid the intended dischaige, and that the enlaigement fiom the 
Inoat to the final section depends imlhei on the back pi ess me 
Against which the sticam it. to escape. 

At any place m the nozzle the dischaige per unit aiea of crost>- 
cction is M v 

~X = V ' 

\.t the thioal, whcie the cross-section is least, this is a maximum 
Consider now the ideal case of isrnliopic expansion in a nozzle 
\ r hen the fluid is one foi \\hieh PV" 1 is constant dining such ex- 
lansion Equation (0) is Hum applicable The \docity at any 
icnnt, the picssmc [hcic ha\mg Jallin to P, is 

P ,'"-'" 



V m - 



] 



, p 
>J 1 



ncl Ihe \olume is V = V 1 ( * ]' 



luice loi tiic disehaii>e pel mill aica oi sccluti; at the place \vheie 
I a k piessme is P, \\c ha,\ i> 

/*& 

\l r 

'Ins may be applied to calculate the piopei sec I ion X foi a gn en 
Lschaine M when the, picssmc has fallen fioin the initial picssmc 
'j lo any assigned IOMCI picssmc P Foi the puiposc of designing 
no/zle there aie only two places wheic this calculation has lo be 
lade, namely at the Lhioat, and at the end wheic the fluid escapes 
"cimst the assigned back picssmc. When the throat-section X t 
nd the final section X f have been calculated, a suitable foim foi the 
oz/lc is icadily diawn, any smooth curve will sci\ e foi the con- 
crgcnt entrance, and any conical tapci may be selected foi the 
ivcigent extension fiom the thioat to the end, provided it is 
cithci so abrupt as t6 interfere with stieam-lme flow, nor so 

132 



196 THERMODYNAMICS [en. 

gradual as to make the nozzle unduly long and thcicby introduce 
unnecessaiy fuction 

To calculate the final section X f which will allow Ihc energy of 
the, fluid to be fully utilized by expansion down to the !is.signcd 
back piessme, that pressure is to be taken, foi the value of P in 
Eq (7). To calculate the section at the bliioat the piessme Ihere 
has fiist to be found The piessinc at the I hi oat is dctei mined by 
the consideiation that the dischaigc pci unit of section (ftf/X) is 
theie a maximum. If the expression foi MjX in MCJ (7) is dit'ler- 
entiated with i expect to P/Pi and the dilfcicnluil wiillen equal lo 
zeio, the lesultmg value of PJP t Mill be that Tor which MJX is a 
maximum, in othei woids it will be the value of 1 /*,//-^, where P, 
is the pressure at the tin oat 

Eq (7) ma} be wntten 



__ 

X~ V m-~l F 

The condition foi a maximum is found bv d)ffcieiilialinj> Ihe 
quantity undei the second lool 



m\P 

2 

fiom which 



- 
P l \ 2 J (in -|- 1 / ^ s )' 

Fuithei, b> substituting this m Kq ((v/), we have foi UK- \ eloeity 
at the tin oat 



^V-Wi ("J- 

The volume (pei Ib ) of the fluid at the lhro.il is 



(10) 

By combining tlia>e an equation is oblmucd tor Ihe 
pei unit of ci oss-sec turn at the thioal, 



__ 
X t V t ~\m + l) V (n, | 1)T, 

From this equation the cioss-section al Ihc [1 U()}|1 ls found 
which will give an assigned discharge when Ihe imlial j)iessuie is 
known. The ratio of the cioss-socLion uL any place, where the 



JETS AND TURBINES 



197 



iicssiue is P, lo the cioss-scclion at the thioaf, is readily found 
torn Eq. (7a) . 

//PA- 
Y v (pj ' 

A ^ x ^'_ - (12). 

//Hi V ' 

P\~^T 



'his expression is convenient in dctxi mining the pioper amount 
f enlargement of the nozzle fiom the tin oat to the end when the 
aek pi'cssuic ii> assigned. 



133. Limitation of the Discharge through an Orifice of Given 
ize IL follows iiom these equations that the dibchaige thiough 
given orifice undei a given initial piessuie P x depends only on 
he ci oss-sec I ion at the nan owes b pait of the oiifice, and is mclepen- 
ent of the back piessuie, piovidcd the back picsbinc is not greater 
ban P L as calculated bv Eq (8). By continuing the expansion in 
divcigcnt nozzle uflei the tin oat is passed, the amount of the 
ischnige is not inci cased, but the fluid acquues a gieatei \elocitv 
cfoiciL le.nes the noz/le, because the range of piessme which is 
Ifective I'oi pioducing velocity is increased To put it in another 
r av, we may say that the heat-diop down lo the prcssuio at the 
uoatdctei mines the amount of the discharge, and the remaindci 
f the hcat-diop, which would be wasted if thcic \\Licnodivugent 
vtcnsion of Ihc no/./lc, is uhh/cd in Ihe di\ ugtnt poihon to give 
clchlional vclociU to the escaping si i cam Tins \clocitv is i>i\cn 
i a dclmilc and iist-lul diuthon, \\hcicas il Ihuc \\cic no di\ oigent 
vlciision ol the no/'/lc the fluid allci Icaxing Ihc 
t)7,zlc, would expand laterally, <mcl its parts would 
n qiiuc velocity in directions such lhal no use could 
r made of I lie kinclic oncigy so acquucd 
C'onsidci what Jiapiuns wilh a nozzle such as that PI 
I' /ig 07, which has no chvcigcnt extension Fluid 
expanding Iiom a chambci where the picssiue 
P } into a sjjacc when 1 Ihc pressure is P 2 
ssumc the back picssmc P 2 lo be less than P t as 
ilculatcd by Eq (8) In that case the picssuic m 
ic jet, wheic it leaves the nozzle, will be P t , and the fmthei 
i op of picssuic to P 2 will occui thiough scattciing of the 
icam. The dischaigc is dctei mined by Eq. (11) It is not in- 
eased by any lowenng of the back piesswe P 2 , because any 




198 THERMODYNAMICS [en 

loweimg of P 2 docs not affect the final piessurc in llic nozzle, 
which remains equal to P t Osboine Reynolds' 1 explained Ihc 
appaient anomaly by pointing out that the sticam is then lea vino 
the nozzle with a velocity equal Lo Lhal with which sound (01 any 
wave of extension and compic&sion) is piopagated in Ihc lluid, and 
consequently any i eduction of the pressure P 2 cannot be com- 
municated back against the stream its effects aie nol felt al any 
point within the nozzle The picssnie in Ihc sticam at I he oufice 
theiefore cannot become less, howc's cr low the kick picssmc 
P 2 may be But if P 2 JCi increased so as to exceed P t , the lalcial 
scattering close to the orifice ceases, the velocity is i educed, I lie 
piessure at the orifice then becomes equal to P 2 , the dischaigc is 
i educed, and its amount is to be calculated by willing P 2 foi P 
in Eq (7) or (7 a) 

In applyng these icsults to a nozzle of any foim, the least 
section is to be legaidcd as the thioat if thcic is a divcigent 
extension beyond the least section the amount of I he dischaigc is 
not affected, though the Imal velocity of the stream is mcicastd 
Taking a, nozzle of anv form, and a constant initial picssiire P, , il' 
we icduce the back pressure P, fiom a value \\hicli, Lo begin Milh, 

is just less than P-,, the dischaigc incicascs until P> reaches the 

a 1 

value P l l' J - j . Aftei that, an) r fiutlici icduction ol P, dots 
in -\- l/ 

not mciease the dischaigc But the velocity which Ihc (hud aequnes 
befoie it leaves the nozzle may then be augmented by lowciing 
P> and adding to the divcigent poition of I lie nox/le The- no///Ie 
will be lightly designed when it piovidcs foi ]iist enough expansion 
to make the final piessuic equal lo the back piessme, Ihc |el then 
escapes as a smooth sticam, and the cncigy of expansion is 
utilized to the full. If the nozzle does not cany expansion I'ai 
enough, if in othei woids, the final piessme exceeds I he back 
pressure, eneigy will be wasted by scalleung. If on I lie olhc i hand 
the back picssmc is too high for the noz/lc, so llial llic no//lc 
piovidcs foi moic expansion than can propcilv take ])Iace, vibiti- 
tions aic set up which cause some waste \ m . \\ r c shall now oonsidei 
the application of these general rcsulls to an and to si cam 

134 Application to Air. In applying the above foimulas lo 

* Phil May Match, 1SSG, Collected Papa s, vol IT, p Oil 

f For experiments on tlio effects of no/zlcs M'hich cany oxpansirm ton far, or 
not fai enough, see Stodola's book on the Steam Tuiljino 



7] JETS AND TURBINES 199 

my penn anen t gas, such as an, the index m is y, the ratio of 
he two specific heats (Ait. 25). Its value foi an may be taken as 
L 10 Substituting this numbei in Eq (8) we have, foi a jet of 
in expanding under isentiopic conditions, 

= 528. 

Hence if the jet is being dchveied against a back piessme less 
Llum 5'2SP l a diveigent extension of the nozzle is requned to 
4ivc the f>icatcst possible velocity to the issuing stream, though 
Lhe quantity dclucicd will be the same as that which Mould be 
delivered against a back pressure ol 528P-L If the back piessure 
be mci eased it must exceed 528P X befoie theie is any diminution 
in the dischaigc. 

As a nunieiical example, suppose that air, with an initial piessuie 
D!' 300 pounds persq inch, is dischaiged thiougha comeigent-clivei- 
<>ciil nozzle into the atmosphcie, 01 against a back piessuie of sav 
]5 pounds per sq inch The piessuie at the thioat is 158 -4 and, 
since Liu final inlio of piessuics is one to twenty, the uitio of the 
(mal cioss-scclion to the cioss-scction of the thioat should, by 
](' ( , MO\ ] K i 

1 V " h X f= VO 101 0-0.3.3 19 =0()() 

X r Vo 013S5 - 00588 

This is 1 01 the ideal case of isentiopic expansion Eftects of Inchon 
air disicoaulcd, they Mill be consiclciecl m Vit 140 

135 Application to Steam In applying the geneial equations 
loi iscnliopic expansion to steam, \\c ha\ c to distinguish between 
Hu h pe of expansion uhich occnis in a jet and the type ol expan- 
sion \\hich was LicuLcd of in Ait 78 In that aiticle the expansion 
\\as assumed to be isentiopic (adiabatic), (/> was constant But it 
\\iis also assumed that ill each stage in the expansion the fluid Mas 
m HUM mal cqiiihbiiuni, it theidbic consisted of a mixtiue of 
s.iluialed steam Milh the piopoition of \\atei nccessaiy to keep 
[he entropy constant The expansion dealt with in that aiticle 
nuiy be ckscubcd as the cquilibiium type of adiabatic 01 isentiopic 



It is now recognized thai the eqiuhbimm t\ pe of adiabatic ex- 
pansion docs not occm m Lhc foimation of a steam jet Foi 
icasoiib which will be appaicnt as we proceed, the steam in the jet 
is not a mixtuie of saturated vapom and water it is moie or less 



200 THERMODYNAMICS [CH. 

supeisatuiated \\hen the equilibiium condition would be one oi' 
wetness At any stage of expansion, the steam, instead ol' being 
in the stable state coi responding to its piessme, is in what is 
called a ineta&table state, a state that cannot be permanent in any 
vapour In the metastable state the steam is snpersatuiatcd, it 
may be completely diy, 01 it may have some watei mixed wilh it, 
but necessanly less than there would be in a stable mixture at 
the same piessme In other woids a melaslablc state exists only 
before the propel fraction of the \apoui has become liquid. In 
passing fiom the metastable state to the stable 01 equiJibnum 
state, at the same piessme, pait of the \apoui is condensed, heat 
is accordingly gnen out, the temperature uses, and the en I ropy 
of the fluid as a ^hole is mci eased 

If the steam is supeiheated to begin with, il behaves like a 
gas in the initial stages of the expansion, and its cquilibiium at 
each stage is stable until it ciosses the boundai}' or satuiation hue, 
that is to say, until its temperatuic falls to a \aluc Avhich cone- 
sponds to satuiation at the piessme then i cached IL is only m 
fuithei expansion, bejond that stage, that a metastable condition 
can be pioduced If the steam is initially satmatcd a metastable 
condition is pioduced as soon as expansion begins 

According to Callendai's equation, the adiabalic expansion of 
supeiheated steam follows the kn\ (Ait 78) 

P(V - &) 13 = const, 

wheie b is a small teim repiesentmg the volume of walci at C , 
namelv 016 cub ft per Ib 

The same foimula continues to apply in expansion beyond the 
satuiation line piouded no watei condenses out, Ihat is to s\, 
piovided the metastable condition of supersaluiation is so com- 
plete that the steam icmains quite di} It also applies, midi-i I he 
same pioviso, m the expansion of initially satuialcd steam 

The expenments of C. T R Wilson (ahciirly iclcm-d lo m 
Ait 79) have shown that m the absence of foicign nuclei, such as 
dust paiticles, and of nuclei due to ionualion f , ualci'-vapom 
does not condense when it is suddenly expanded unlil its piessurc 
is laigely i educed, and then a cloud of small Avatcr-paihdes is 
obseived Even then, howevei, the conditions aie not those of 
eqmlibmim, for when the expansion is continued a much denser 
cloud, composed of many moie paiticles, appears at a latci slagc 
* Wilson, Phil Trans A, vo] 192, p 403, and vol IDS, p 28') 



] JETS AND TURBINES 201 

Wilson's experiments weic made by expanding an (01 oilier gas) 
i tin a ted with Avater- vapoui, but tlie geneial conclusion would no 
oubL appty if water- vapoui were expanded alone 

Given plenty of time, a condition of equilibimm \\oulcl be 
Cached by condensation of pait of the vapoui on the walls of 
ic containing vessel, but in the veiy lapid expansion which occui> 
uring the passage of steam through a nozzle, condensation on the 
mcr sin face of the nozzle can do hltle to^aids bunging it about 
he effects of sin face condensation aie insignificant Hence in the 
irher stages of the expansion, as fai as the thioat and for some 
ay beyond it, steam behaves like the vapoui v m Wilson's evpeii- 
icnts befoie the clouel of watei -particles appeal eel, it is supei- 
ihnatccl anel puictically diy This is true of steam that is initially 
ttuiatcd, anel afo)twn of steam that is initially supeiheated 

It follows, as Callendai has pointed out 1 , 'that in calculating the 

schaige of steam thiough a nozzle with a gnen size of thioat, 

the si/c of thioat requiiccl foi an assigned amount of discharge, 

ic piopci foimula lo use, in the ideal case of isentropic expansion, 

that which lefeis lo super^atuiated, as well as supuhcated, 

earn, namely p ^y _ ^ j = const 

lie l<im b is lehitivclv so smnll (e\ee[)l al \ci\ ln<h pussuu) 
ial il ma\ as a mle be ncgloeled, m which case the equations 
iead> gi\en will appl\ wilh Ihc \aluc 1 3 foi Ihc index m 
Thus if \\e lake the loimula as PV 1 l = constant (omitting b) 
id apply il in Equation (S) lo find I he piessine al the tin oat, we 

\,\ e p 

' = 05157 
* i 

I! tic-count wue taken of the Uim b C'allendar| sho\\s lh.it this 
piession would bec'Oiue 

P b 

1 0-51.57 130 

' i ''] 

ic- sm.ill lei in chptudmy on b amounts to less than 0001 \\hcn 
e unlial pussuie is even iis lugh as 200 jiounds pci sq inch it 
;>y, therefotv, be omitted m any piaclical calculation of /Vind ^ c> 
ay lake 515 us Ihc ratio of thioat piessuie lo initial picssme 
i a .steam ]e-l Tins applies whcthei the si earn is salmated 01 
jjcrhealed to begin with, m eithci case the steam is diy when it 

" "On tho steady How of Htoaiu tluough a noz/lo 01 Uuottlo " Pioc Jnst Mech 
g , Fob 1915 

- IjOC Clt (1 t 



202 THERMODYNAMICS [CH. 

passes the thioat, and it will be supersaturated thcie unless theic 
has been much initial snpeiheat. 

Befoic it was iccognized that a jet of initially saturated slcam 
is necessanly snpcisaLiuatcd when it passes, the LluoaL, it was 
usual to calculate the thioat pressure by taking, for the index m, a 
value appropriate to the equihbimm type of acliabalic expansion. 
The index 1 135 was gencially taken as applicable ! , with the result 
of making P t = 577 P^ The equihbimm theory, which is cei- 
tamly inconect, thcicioic made the picssuie in the lliroal Loo high, 
consequentlj r the calculated dischaigc for a given size of throat 
was too smalJ Experiments on the flow of steam thiough no/zlcs 
weie then found to give a dischaige winch was actually gicalci 
than that which had been calculated foi the ideal case of no 
friction, although the effect of fnction would be to make the 
actual dischaige loss than the ideal discharge When, howcvci, 
account is taken of the fact of supeisatuiation, by using 1 3 as 
the index, the calculated discharge is biought into haimonv with 
the icsnlts of experiment. The ie\iscd thcoiy gives a calculated 
dischaige shghLly greater than the actual discharge, but with no 
mote difleicnce than can piopeily be ascnbecl to fnchon 

Using the index 13 m Eq. (11) we lia\ c, loi the dischaige 
per unit aiea of section at the tin oat, 

10 . , 

M 2 \ a /2 \ 322x13 /P L 



X t '23; V 23 V F/ 

- 3 780 k /?i 
V TV 

with pounds and leet as units throughout With the units moic 
commonly cmplo\cd this gives 

A/ m Ib. pei sec. _ /P l in pounds pci sq inch 

X t in sq inches 'V F T in cub. ft pei II) ' 

as a foimula foi calculating the sr/c of throat in a no/zle that is 
supplied with cithci satuiatcd 01 supcihcatcd steam On the 
equihbimm theoiy the numciical factor was 03003 instead ol 
03155 The collected theoiv makes the chscluu i>e about 5 pei 
cent gicatei. 

Aftei passing the throat some condensation no doubt oceuis m 
the form of a cloud of .small watei-pailiclcs, as in Wilson's cxpcii- 
mcnts. But the piocess takes time, and the whole time occupied 

* Compare, for example, the authoi's book on The Slcam Engine and othei Ileal- 
ti, Edition of 1910, p 214 



] JETS AND TURBINES 203 

V the steam in passing through the nozzle is so shoit that it may 
e doubted uhether the condensation that occms within the nozzle 
ocs much to icstorc equilibrium 'It is probable that the steam is 
.ill to a large extent supei satin a ted when it escapes*. As icgards 
alculalion of I he final aiea of cioss-section no gicat error will 
c mtiodnccd if A\C considei the formula PF 1 3 to be applicable 
noughoiiL, and this founula will also give a good approximation 
\ estimating the final velocity of Lhc jet 

136. Comparison of Metastable Expansion with Equili- 
num Expansion It may help to make this mattci intelligible 

we compare move full}- the adiabalic expansion ol steam under 
icli conditions that it is a \\ct mixtmc in a state oJ equilibrium 
uouiihoitt, M'llh its adiabatic expansion m a melastable state, in 
'Inch it icniams completely chy Let steam expand from an 



Volume 

F\s, (>S 

nti.il slaLe icpiescntcd by a (hi> G.s), m \\hich \\e \\ill assume it 
i bi- di} and sahnated Tlic cnnc m icpiesciils adiabahc c\- 
aiision ol Ihe h pc licalcd in Chapleis II and III Al t-.uh slayc 
I Lli.il pun ess M\c Mind is 21 mivliiie, m slal>lc equilibi mm, of 
ilmalcd vapom audwalei Us \ olmue at an\ picssinc is de-lc i- 
imed l>> 1hc inclhodcxplamccl in Vi I 7S The cmvc ab icpiesenls 
lelasl.iblc- adiabalic e\[)ansiou dining which Ihe sLcdm remains 
mlc div. Us lonn is dele i mined I)\ Ihe equation 

p () _^)i ' = const | 
n both cases the expansion is isenliopic and theieloie 




* Olisi-i uilions ol tho appoaumco of escaping jets mippoiL HUH ccmc-Iiision They 
io\v lliaL wlitn steam iiuli.illy diy (but not nocoss.mlv supoilicatod) escapes fioni 
divoi"cml nn//lo in whu'h it lias cvpandod LluouKhaconsidoiablGratio, nopaitjch^s 
watui bocomo visiblo until tho steam haa tiavcllod some difitanco from tlio 
idco you lytodola, Zciltithnft dcs Vacuum deuLi,Jier Inrji>/iieu>e, 1913 
f Hoio, and on p l!04, the b ot Callondai's equation is written /3, to avoid con 
ision with the b of tho diagram 



204 THERMODYNAMICS [en. 

But though the eutiopy and the piessiuc aic the -same at b as at c, 
the fluid is in two veiy dil'leient states. At b it is a homogeneous gas , 
at c it is a wet mixtuie At c its tempeiatuie is the Icmpciature of 
satuiation coiiesponding to its piessme theic, at b its temperature 
is much lower, being detei mined by the equation P (V /3) = KT, 

3^ 

which makes T/P^ = const. The volume is of course less at b. 

The heat-diop fiom a to c is the theimal equivalcnl of the woik 
lepresented by the area eacf, and the hcal-diop fiom a Lo b is I he 
theimal equivalent of the work lepiesented by cctbf, since bolh 
types of expansion aie adiabatic (see Ait. SO) Ilcncc the heat- 
drop is less in the metastable expansion, by an amount that is 
eqimalent to the aiea abc, and the total heat at b is theieforc 
greatei than the total heat at c by that amount 

The total heat I of the mixtuie at c, aftci equihbiium expansion, 
may be found by the method descubed in Ait 89 01 Arl. 1)0. 

The total heat 7 6 in the metastable fluid at b may bo iound by 
leckomng the heat-drop from the initial value / Since 



the volume at any stage in the metastable expansion is 

m 



Then, since 7 a - / - A (Aien eabj ) = A\ VdP, 

] P>, 

ID ,P lt 10 ,/' 

7 fl - I, = AP a i* (V a -j9) <IP/P" | A? <ll> 
1 PI, ' i\ 



Suppose now that after sudden expansion lo b, <ik>nn (he cm\o' 
ab, the metastable fluid at b is allowed to become slnblc by pai I i.iJIy 
condensing undei constant piessure, without any ain or Joss of 
heat Its temperatuie will use to the saliualion value lor Ihal 
piessme, it will, theiefoie, come to have the same lemperaliiie 
as the mixtiue at c, but it will be somewhat dnei, because its total 
heat lemams equal to I b which, as ^e have seen, is gieatei Uiaii 
the total heat I c of the mixture at c Its volume will, Ihoicfoie, 
mciease to a point d, \\hicli is beyond c 

If we wnte q c foi the cli j ness at c ol steam that has exj)anded m 
a stable state, 01 state of equihbiium as a whole, fiom a Lo c, and 



] JETS AND TURBINES 205 

{ for the diyiiess at d of steam that has expanded in a metastable 
ate to b and has subsequently attained equilibiium, by water 
paratmg out at constant pressuie, without loss or gain of heat, 
ic diffeiencc of total heats is 

I d - 1 - L (q d - q c ) 

ut I d = /(, and / & = I c + A (Aiea abc) 

[cnce L (q d q c ) = A (Area abc) 

In attaining cqiulihiium the fluid as a Avhole has gained entiop3 r , 
>r (j> d is gicatei than (j) b , 01 ^> a) 01 </> c in the equilibiium state, by 
le amount that woujd convert the equilibiium mixtme at c into 
ic equilibiium nuxluic aL d. Thus 

, , _ L (q d - q c ) _ A (Area abc) 

<Pd ~~ <P& rn rfi 

* d J- d 

Jus mcieasc of entropy is not due to any gam of heat, foi no heat 
as been communicated to the fluid, it is due to the fact that theie 
as been an nie\ oisible internal change in passing fiom the meta- 
ablc to the stable stale 

We may tJunk of the substance as midcigomg a cycle of changes 
lailmg fiom a let it expand suddenly and adiabatically to b, 
icn Jet il change fiom b to d at cons I ant pi ess me without taking 
i 01 gi\mg out heat Then let it be partially condensed, undei 
msl.int picssme, fiom d to c, duimg this stage a quantit\ of heat 
msl be gueii out equal to L (q d q c ) Then lei il be skmlv com- 
icsscd along Ihe equilibiium adiabatic curve fiom t to a. llus 
impletes tlic c> cJc VVoilc has bcui expended, equal to the aiea 
'jc, and a coiicspondmg quantity of heat has been ic,mo\ed 

Duimg its tiansition (along bd] fiom the mclaslablc to the 
abJt state, the fluid passes tlnough a state in which its picssuie 
id \olume are the same as those of the equilibiium mixtme at 
But its slate in othci icspccts is by no means the same, it is 
icn a mixtuie ol supeistituiatcd vapour wilh some liquid, not in 
uiilibiium, its tcmpc'iatuic is lo\vei and its total heat is gieatei 

It is scaiccly necessaiy to add that the rcmailcs which MCIC made 

Ail. 75 about the specification of the state of a fluid assumed 
tat the fluid as a whole was in equilibrium They do not apply 
metastable or tiansition states 

As a numerical illustiation of the comparison made in this 
tide, assume dry satmated steam at 100 pounds pei squaie inch 
. expand adiabatically to 35 pounds. In the initial state the total 



206 THERMODYNAMICS [on. 

heat is 661 82, the tempeiatuie is 161 28 C , and the' enhopy is 
1 6082. 

Suppose fiist that the expansion occuis without niy \\alei 
sepaiatmg out We then have 

Heat-chop l a - I b = 42 02, / & = 019 20 

The tempeiatuie falls fiom 164 28 C. Lo 70 2 C 

The volume at b is 9 962 cubic feet. 

If, aftei this expansion, the metas table vapour a II. mis eqm- 
libiium nuclei the constant piessiuc of 35 pounds pc'i si|iuie inch, 
without gaining or losing heat, it changes into wel sham at a 
tempeiature of 126 25 C \\ith a diyness q d = 9 123 and .1 volume 
V d = 11 210 cubic feet Its enliopy inci cases (o 1 01 I-U. 

Suppose, on the othei hand, that adiabatic expansion lioiu I he 
initial state at a occuis along the cquihbiium cmvc (. \Ve I hen 
have ^ = 616 g0j heat _ dlop / fl _ / t = j/5 oa 

At the end of the expansion the tempeiatuie has fallc-n only In 
126 25, and the steam is a wet mixtuie \\ii\\ diyncss f/ t - 'KT77 
and volume F = 11 155 

The aiea abc, which rcpicscnts work lhat is lost in I lie Insl 
method of expansion, is cqui\alent to the dilleiencc lu-l \uui llu 
two heat-diops, namclj 2 40 Iheimal units The loss u hich icsnlls 
fiom supeisatination is thcitloie ncail\ six pel cuil ol I lie ,\\ a li- 
able heat-diop This loss of available cucii>v, \\lueli <<cms in a 
nozzle as a result of supeisatuiation, is distinct horn and addi- 
tional to any loss that may occm tliioiigh fuel ion 

137 Measure of Supersaturation hupcisahnal u>ri m\ol\is 
supetcoolmg, that is to say, the vapom is cooled In low llx 
satuiation point coiiespondnig to its pu-ssuie 1 In UK .il>o\( 
example the supeicooling at b is 50, nanulv 120 2 - 7() y Snpi i- 
satuiation also involves an excess of picissmc, and a COM < spondmn 
excess of density, when companson is made bc-hvein llu> |>iessme 
01 the density of a supeisatmatcd vapoui and llial ol ,1 .salmahd 
\apoui at the same temperatuic Thus at b m the ix.iiuple the 
piessure of the supei satin ated vapoui is ,'35 pounds, wlu-ivas llu 
piessme of satmated steam at the same tempciatme is only 4 fi/i 
pounds The latio of densities is neaily the same as I he ratio ol 



* This is called ' undoicoolmg" by some Miitcjs, but, tlio won! ',.,, ,,,,l,m'" 
is moie appropriate as a do^nption of cooling wluch w in OVCSH ,,| ll,n 
amount The expanding vapoui is cooled loo much, not too IiLLlo 



1 JETS AND TURBINES 207 

L-essures Supersaturated vapour behaves like a gas with PV 
early constant at constant tempeiatme Either ratio selves as 
convenient means of specifying the degiee of supei saturation. 

138. Retarded Condensation. Wilson found that when clust- 
ee air, satuiated with watei- vapour at 20 C., was adiabatic- 
lly expanded by suddenly enlaigmg the vessel in which it was 
mtained, a cloud of fog paiticles did not form until the volume 
L' the vessel was nici eased in the latio of 1 375 to 1 This, coire- 
Donds Lo a ncaily eighL-lold supeisatuiation of the vapour, that 
to say the ratio of Lhe vapoui densities, 01 of the actual vapoui - 
lessuie after expansion to the piessuie of satuiated vapour at 
ic temperature i cached by the expansion, was then neaily 8 
[e J'ound that the time-rate of the expansion might be vaned 
msideiably without affecting this result, and also that when the 
vpansion was cained fiuthei a much dcnsei log cloud was 
)imcd, containing man}' moie paiticles It followb fiom these 
suits Lhdt the gio\\th of those fog particles which MCIC mst 
umcd did not go on fast enough to icstoie and maintain cqui- 
biiLim in the expanding fluid 

Condensation of expanding steam, by the foimation of \\atei 
aiticles suspended in the Aapom, is uccoiehnglv ictaidcd in t\\o 
ilfeu'iit wajs Tlieic is uhat ma} be called a static letaidation 
Inch docs nol depend on the limc-ialc ol expansion, toi the- fog 
oes not begin to J'oim until the A ohunc has incieased by a definite 
nd consieleidble amount In addition, Ihcie is <i time-lag which 
icvenlb cquilibiium horn being i cache el \\lnle 1 tlic expansion 
jiitmues One icason loi this is lhal Ihc chops, once Lhc^ ha\ c 
)iincel, must ha\ c tune Lo cool in oielei thai Uic\ ma} continue 
^ act as cutties foi condensation Ilcncc the menc lapid the 
vpansion the lest, neai will be I lie appioach Lo eejuilibiium at any 
age. altci condensation has begun. 

It may be emcstioncd \vhcthci, even in such slow expansion as 
ccurs m stcam-cngincs oJ the ^hnclci and piston U pe% cqui- 
biiinn of the woikmg fluid is appioxnnatcl}/ attained, noUulh- 
Laneling Lhe assistance which is gn r cn by condensation on the 
ictal surfaces. It is quite possible that exhaust steam discharged 
3 the condensci ma> consist in pait of supci cooled vapoui 
lough it also contains water*. Supersatmation in it would be 
* Compaio Callondai and Nicholson, k On the Law of Condensation oi (Steam," 
hn PIOL In&t C E vol xxxr, pp 171-174, whoic expenuioiilnl ovidenco ib 
icnlioncd of uiipoi&atiuation during expansion and exhaust. 



208 THERMODYNAMICS [c-u. 

readuy detected if we could observe the temperature and eompaie 
that with the piessurc, but attempts lo mcasuie the Umpeialme 
of supeisatuiatcd steam directly, by means of a Ihennomcli i, 
oive fallacious lesnlls on account of condensation of -\\alir on 

Q 

the bulb, 01 on the pocket m winch it is enclosed, 01 on I he 
wne if it is an exposed LheimomeUi of the lesislanee lype. The 
theoiy ol ideal steam-engines using adiabahe expansion, \\lueh 
was discussed in Chaptei III, and from which ellicicneies ol I he 
"Rankine Cycle" wcic calculated, assumed a condition of <qui- 
hbrmm on the paib of the woikuig llmd Ihioughonl I he whole 
opeiation. So far as there is any departure' Irom thai eondilion 
m a ical engine iL makes for reduced clheieney in I Ins ,is well .is 
m othei icspects the real perfomiance ol an engine falls shoil of 
the standaid set by the llankinc Cycle. 

In the more lapid expansion which steam undeigots while il 
passes tluough a tin bine of any lype, U appejus lhal Ihesl.ih is 
far from being one of equilibrium e\ en in the lalei sl.igrs This 
view is suppoitcd by an examination ol I he icsulls of I mils ol UK 
pei foimancc of tin bines, woiking muUi v.uious condilions .is lo 
exhaust presume and initial superheat '. 

The icason tvhy diops ol h<pud do not loim h<(l\ enough | ( , 
pievcnt an expanding v.ipom fiom becoming SIIJK i IK ,iU d \\ill In- 
dealt with moie full} in Vppcndix I II \\ill In <\pl.im<d HUM 
that the static ictaidation icfcned lo m Ihis aihele oceuis .is .m 
effect of suilacc tension in the lupnd 

139. Action of Steam m & Nozzle, continued K< 
now to the action of s team in a 
nozzle, \\e nuy note in passing 
howmetastable expansion mny 
be lepiescntcd on the eiiUopy- 
tempeiatmc diagiam, 01 on 
the Mollier diagiam ol cnliopy 
and total heat. 

Taking fhsL the enliopy- 
tempeiatuie diagram (fig. 09), 
adiabatic expansion fiom an 
initial state a wheic Lhc pics- 

sme is P^Lo any lower piessmeP 2 , under e(iuihbnumcomhliojis, is 
icpiesented by the iscntropic ac, where r, is on the equilibrium hue 

* See H M Martin, "A Now Thooiy of tho NLoiun 'J'uilund, 1 
vol cvi 1918 




JETS AND TURBINES 



209 



>f constant piessme for a wet mixtme at P 2 But if the expansion 

5 so sudden as to occui without condensation, it is repiesented by 

he isenLropic ab, wheie b is a point on the constant-pressuiecuive 

// for supcisatmatcd vapoui. That curve is a continuation, below 

he boundaiy line ah, of the consLuit-pressinc cuive for supei- 

caled steam at P^ TJic ultimate state d which would be reached 

cquilibimm wcie attained at constant / and constant P (as in 

g 08, Ait. 130), may be calculated, but it would seive no useful 

in pose to attempt to lepicscnt on this diagiam the ineveisible 

ansition fiom state b to state d The diagiam shows clcaily 

ic amount of supci cooling cb 

In the /< diagiam (iig. 70) adiabatic expansion under equihbnum 
jmditions is again icpiescnted by the 
entropic line ac, the straight line ch 
2ing the eqinhbiium line of constant 
L'essuic I'oi a wet mixtiue at pi Cb sine 
, Adiabatu' expansion under diy 
ipeisiitinated conditions is icpie- 
nted IJN ab, b is ayain a point o\\ 
ic- constant-piessuie em\e bh ICM 
pcisalmalcd \.ipom, uhich is a 
nlinualiou below the boiiiidai}^ 
i\c ah (A the t'onslanl-piessuie 
i\e loi siipc-i heated steam al P., 
.it' \YL m-i\ dettimmc d oiapluc<ill\ 
ch.i\\ini> <i hoii/onlal stuunliL hue 
loiinh b lo meet Hie upuhbi mm 
nsl.mt-piessine line m (/, I he as- 
umlion being, as bdoie, lli.il the 
l.istabk vai>our, al'lti expansion, 
nuaUK eonus lo <i sl.ible slale in d 
Ihonl ehanuic ol piessuu- and with- 
t i>ain 01 loss ol heat Tin houzontal 
<u<>ht hue !>d is a line of constant 
al ht.aL 

tn UK so diagiams, as well as in Iig 68, we have assumed that 
' si earn is satmaled to begin with But the consti action can 
piously be modified to apply to steam with initial superheat, 
1 point a may be anywhcie in the constant-jnesbuic line for Pj, 
L^O Effects of Friction The losses that occui in jets 01 
bines thiongh friction and thiough snpeisatuiation cannot be 




70 




210 THERMODYNAMICS [en. 

sepaiatcd in piadicc, but foi the pmpose of considumg the 
thei nvxh nannc effect of faction it will bo convenient to Ircat 
that separately by imagining a case in which thcie is no supoi- 
satination Such a case may he icahzed by using steam that is 
highly supeihcated befoie expansion 

Let BC be the eqnilibaum achabatic curve on the piessuic- 
\olunie diagiam diawn, say, 
foi 1 Ib. of steam: then the 
aica ABCD (fig 71) repie- 
sents the amount of woik 
available foi setting the steam D 
m motion as a jet, or foi C c 

getting mechanical clfect out ^ Ig 71 

of it m any mannei The aiea ABCD is cqimalent to the whole 
heat-chop in adiabatic expansion nuclei eqnilibaum condilJons, 
and measmes the utmost woik obtainable in any mclhod ot 
utilizing the eneigy of the steam It is on this basis thai the woik 
of the Rankine C}de (Ait 87) is calculated, which loans an ideal 
stanclaid with which the actual output ot any steam-engine 01 
steam turbine ma\ be compaicd The actual output pci Ib ol 
steam is ncccssaiih less m all cases than the aica ABCD, and the 
latio ol the actual output to that ana is calleel the "elheieney 
latio" of the engine 01 tuibme(Ait 91). In a steam jet the output 
is the kinetic eneigy of the jet itsclt. 

Considei now the eflect of faction m a ne>7/le Assume the 
conditions to be adiathcimal If there \\eie no (action (as well ,is 
no supcisatmalion) the v\hole woik icpiescnted by ABCD woiilel 
be utilized in giving \elocity to the jet and would appeal m il us 
kinetic encigy in that case we should luu e 

V ~ a = Aiea ABCD -- 7 X - /, , 

to 

where v is the velocity produced m the jet (slai ling dom icst) and 
/! / 2 is (in units of woik) Lhe heat-chop m adiabahc expansion 

But in any ical noz/le thcie is some lad ion between Ihe Ihud 
and the sides of the channel., and the flow is lo some exlenl tui- 
bulent, which means that eddies aio formed in which llicic is 
dissipation of the eneigy of flow Lhiough internal i act ion We 
shall apply the word faction bioadly Lo all such losses Then 
elfect is as follows On dibchaige, or at any stage during the ex- 
pansion, the jet has less kinetic eneigy, and thciciorc less velocity, 
than it would have at the same stage if there were no Inction. 



v] JETS AND TURBINES 211 

But its volume, after expansion to any given pressme, is gieater 
Lhan it would be if there weic no fnction, because the eneigy that 
has been dissipated thiough fuction has taken the foira of heat. 
Thus up to any stage in the expansion there has been a loss of 
kinetic cncigy, but theie has also been a gain of heat. Conse- 
qucntlv the fluid has a gi cater volume than it would have in the 
absence of fuction Moreover it has a gieater stock of heat still 
available foi conversion into work in the latei stages of expansion, 
though that advantage cannot in any event compensate completely 
foi the loss ol eneigy to which the increased stock of heat is clue 
The heat that is icstoied at any stage as a lesult ol fuction has 
losl availability foi conversion into a mechanical foi in, foi the 
woikmg substance then has a lowei tempeiatiuc than it had in 
the caihcr slages when the mechanical eneigy was gcneiated out 
of winch that heat has been pioduced Thus the net icsult is to 
ml uce Lhe kinetic eneigy of the jet belo\\ the standaul foi no 
i'liclion, although part of the eneig\ thai has been losl thiough 
I'nction up to any stage is lecovcicd in subsequent stages 

The mattci may be put in anolhc'i way b) saying thai, in con- 
sequence oi fnction, the Iliad, allei expansion to anv piessuie, has 
snllcied less diop ol total heal than it would ha\ e sufkicd had 
Ihcie been no liiclion Theie is less mechanical cilecl, but theie 
is moic IK at kit in it and its \olunic is gicatei, at each stage 

A j)iogicssivc mcicasc ol cnhop\ occuis dining expansion, as a 
lesnll of I ho uie\ eisible pioccsscs IhaL aie i>omg ou \\ilhm the ilmd, 
whcicas with no lnction Hie cnliop^ \\ould I)C cunslanl 

Taking I he picssin o\ olume diayiam, liy 71, I he el lee I ol Inchon 
is lo give the actual expansion cmvc a loim such as LtC' , in \\hich 
the fluid has a gicalei \ oliiino at each slaije than it \\ould ha\ cm 
tht 1 adiabalic piooiss topic-suited b} UC Bui though this ap- 
paicnlly implies a gam of woik Iheie is ically a loss The aua 
AHC'D docs nol mcasine an ac'tual output ol woik, but an 
ailihcial quanlil} which we ma\ call Ihc "gioss appaicnt woik " 
Ol this <>ioss appau'iil \\oik, a paiL is icconvci ted into heal, as 
I he expansion pioceeds, namelv a quantity sufficient to supply 
enough heal at each stage to bung the expansion cm\c out liom 
liC to JJC'. At 1 1 ic end ol Lhe opcialion the net amount oi woik 
thai is obtained, far horn being gieatei than the adiabalic aica 
AJJCD, is less than that aiea by the equivalent of I z ' I 2 , wheie 
/ 2 ' is the total heat at 6" and / 2 is the total heat at C In 
other words it is less by the quantity of heat which would be 

142 



212 THERMODYNAMICS [en 

required to change the condition of the expanded llmd at constant 
piessuie from C to C' . 

To piove Lhis we may think of what happens when the substance 
is cained through an imaginary cycle. Staiting fiom stale B let 
it expand, with faction, to C' Let W be the net amount of uoik 
actually done by it in this expansion. Then let iL be changed from 
state C' to state C by lemoval of heat nuclei constant piessuie. 
The quantity of heat so icmoved is /,' - /,. Then let it be com- 
piessed adiabatically fiom C to B The work W done upon it 
dm ing the compression (Avhich is reveisiblc) is the same as the 





C C' G D 




MM' MM' 

Fig 72 Fig 73 

work that would be done by it in adiabahc expansion The c^elc 
is now complete, and by the consciyation ol energy we have 

W + (I/ - /,) - W - o, 
01 W = W - (// - I 3 ) 

Hence also W = I 1 -I Z - (/,' - /,) = /, - //, 
or the net amount of woik done it, equivalent lo the ai'liwl heal- 
diop, in agreement with Art. 10-1.. 

Tmnmg to the entiopy-tempeiatme chagiam (lig 72), the ideal 
case, without faction, is lepiesentecl by ABCD Friction gives 
the expansion cuivc some such foim as BC' ', in which the entiopy 
increases piogiessively as the tempeiatme falls The aiea MBC'M' 
rcpiesents the heat produced by faction, it is the heat icqmicd to 



JETS AND TURBINES 



213 



give the expansion curve its actual form, and since no heat comes 
from outside sources it is supplied at the expense of the kinetic 
cncigy of the jet, by a conversion which is going on horn the 
beginning to the end of the expansion The gioss apparent work 
is represented 11 by the area DABC', but fiom this. \\c have to deduct 
the area MBC'M' to lind the net amount of woik which finally 
appears as kmelic cncigy in the jet Thus allowing for faction the 
net amount of work ]]" is 

Aiea DABC' - Aiea MBC'M', 
01 Area DABC - Area MCC'M' 

Hence the net loss, as compaied with the work \\ that would be got 
in achabalic expansion (with no fncliou), is the area MCC'M', 
which is /,' 1 }> as f^^ e 

In lig 7'2 the steam is initially satin ated If it be. supei heated, 
let B be the initial condition 
(fig 73), AEB being the con- 
stanl-picssiiu line foi I\ anel 
DCC' the const, inl-piessinc 
line I'oi P, Fi ichonlcss ex- 
pansion (in cquilibunm) 
\\unld be H piLsenled b\ BC 
Die <ie tnal e \pausion is along 
somi sue h line 1 us IiC' The 
gioss ,ippai i nl uoi k is H pi e- 
si nil d \)\ I hi 1 a i ea /) ittBC'd, 
,iii(l I hi IK I .unoim! is loiind 
l>\ di dm ling Iioni I h.il the ~ 
,111 .1 M BC' I/', \\ Inch 1 1 pi e v - 
si ills I hi he <il di"\ i loped I hi o ugh Inchon 
is lo deducl an amount ol woik equal lo lhe aiea MCGC' M' from 
I he ulial pel foi manee D itittC This eleduclion is equivalent to 
// /._, as be I'oi e 

Foi piacheal pui|)oses il is moie useful to upiescnt the ellccts 
of Inelion oil the Molliei diagram ol tnliop's and total heal (lig 7 !) 
Le-l B icpiesent the initial state (in this example theic is some 
superheat, Ihe 1 bioken line is lire bomulai} eiine) BC repicsents 
an ideal adiabalic process ol expansion and BC' the actual pioccss 
/T is the initial total heat, / 2 the total heat thot would be left in 
Ihe steam alter adiabalic expansion lo P 2 , and /,' is the total heat 
actually lefl in the steam after expansion to that pressure The 
* Subject to Uio small corieoliou mentioned in Art 88 




Entropy 
ITi},' 71 
I'he net e fleet of diction 



'21 i THERMODYNAMICS [en. 

actual heat-diop, to which the net amount of work done is 
equivalent, is BK or /j - / 2 ', and the net loss resulting fiom fnction 
is KC 01 I z ' - I 2 

When the piopoition is known befoiehand of the friclional Joss 
KCto the total theoieticallv available hcaL-diop BC we can maik 
the point K on the achabatic line through B and diaw a homontal 
line thiough K to find C' . When Lhcic aic ex pen men I a I data i'oi 
estimating the fnctional losses in expansion clown lo vtuious inlci- 
mediatepiessmes we can apply this construclion lo fi.ice Lhe actual 
expansion cuive BC' in a senes of steps The method is applied 
to compound steam tin bines, as a means ol determining Lhe stale 
of the steam aftei each of a sencs of stages in the passage oL' steam 
thiough the tin bine (Ait 1 15) 

The student may find it useful to cxpicss I he effccl of fuclion 
thus When theie is no fuction, and the expansion is adiabalie 
(Ait. SO), (U = y dPj 

where dl lepiescnts (in units of woik) tlic diop ol tola! heal 
which takes place while the piessme diops b) dP. When Lluic i^ 
friction Al > = y d p _ cl Q^ 

\vheie dl' icpiesents the drop of total heal as affeeled b\ 1 1 id ion, 
and V the volume as affected by faction, dQ. being UK qiianhlv 
of heat geneiated by fucLioii at the expense ol Hie gioss ,ipp,iu nl 
work and restoicd to the fluid as heal lie nee 

dl- dl 1 = (1Q- (V - V)dP 
Integiatmg between the limits (i) and (2) 



wheie Q is the whole quantity of heal geneialed l>y Ii 
Since 7 X and 1^ aic the same, tins gi\es 

I z ' -I 2 = q- Aica BCC' of fig 71, 

which expiesses the fact that in consequence of fncLion I he nel 
loss of mechanical effect is equal to the heat geneialed, less I he 
woik that is iccovered thiough the augmcnfalion of volume 
which fnction bungs about. 

141 Application to Turbines The above discussion ol the 
effects of fnction lelales, in the fiist phicc, to ]els, bul it may 
icadily be extended to tuibmes When a jet is eonsideicd alone 
the "efficiencv-iatio" (Ait 94) of the piocess, rcgaidcd as a eon- 
veibion ol eneigy, is measuied by compaiing the actual kinetic 



v] JETS AND TURBINES 215 

eneigy of the issuing stream with the eneigy obtainable undei 
ideal conditions, that is to say with the adiabatic heat-drop. But 
in a turbine, of any type, the piocess of conveision goes fuithei 
the object is to do mechanical woik on the lotor or revolving pait 
of the machine, and the efficiency of the piocess ax a whole is 
measmed by comparing the work done by the fluid on the rotor 
Avith the acliabalic heat -di op. 

142. Simple Turbines In one type of tin bine (De La\ al's) the 
two opeiations aie enliiely dislmcL The steam enters the tmbine 
thiough a fixed nozzle of the convcigent-diveigent type (fig. 66) 
in which Us piessme diops in a single step thiough the \vhole 
a% tillable range fiom the initial piessure P x to the back picssure 
P, Subject lo fuctional losses all the eneigy takes a kinetic foim 
in the ]cL The jel I hen impinges on blades that pioject fiom the 
cncumfciencc of a vciy lapidl} ie\ol\mg wheel, acting on them 
just as a jet ol wulci acts on the blades of a Pelton watci-uhcel, 
\\ith an impulse which is mcasuied b\ the loss of momentum of 
Lhc sticam The ]el theiebj convcits its kinetic cnfiyy into woik 
done on the wheel This second comcision is a pinch l^diauhc 
])ioccss, a question ol d\nanncs but not of thei moch naniics It 
imolvcs fiiclional losses (dislinct 1'iom the eaihei hietional losses 
ID I he no/7le) a^ \\cll as losses aiismg liom the lad that the stieam 
is not \\holl\ ckpiixccl of momentum In its impact on and passage 
mil the K-\ ol\ ing blades The eThcii'ii^ -latio of the \\hok piocess 
is theiel'oic ddeimimcl as the pioilnct ol \.\\o l<idois, naiueh, 
the clliciuic'\ r -]<ilio of coin eision ol UK steam's hcal-t m iy\ into 
eiKi<>\ ol motion on Hie pait of UK ]ct, and the ialio m ^Inch 
lh.it encigv is atlci wauls coiutited into ssoik on UK ioloi 

II is only m wh.iL aie called ^tm^lc Unbincs that these lactois 
aie iiulependent, nainth, 1 in bines m ^huh the vdiole expansion 
occuis in a single step bcloie, the action on the blades begins In 
some such I ui bines the lotoi has mote than one ung ot moMiig 
l)hules, and between successive imgs ol moving blades thcic an 
nngs of hxcd ginclc-bl.ieles which altci the elncction of the stieam 
but do not contribute any addition of kinetic eneigy, loi thcie is 
no expansion m them Tuibincs of this kind aie still ' simple" 
in I he thcrmodynamic sense, for the whole prcssiue-chop is com- 
plclc belore the momentum of the sticam is utilized, and m all 
such turbines it is possible to distinguish clcaily between the two 
lactois. 




210 THERMODYNAMICS [cir. 

143. Compound Turbines Most tin bines, howevci, aic com- 
pound, the expansion takes place in a sencs of steps or stages in 
each of which woik is done on the lotoi Each stage uses only a 
fiaction of the whole heat-chop, leaving the icmaindci Lo be used 
in latei stages In each stage thoie is a convcision of pait of Lhc 
steam's heat-energy into work and theie is fncLional loss both in 
the nozzle and the blades The heat pioduecd by Lluil loss augments 
the quantity of total heat which the steam caines on Lo the next 
stage, theie is, theiefoic, in the subsequent stages a iccoAciy of 
pait of the loss When the stages aic \ciy nnincions, is they arc 
foi instance in a Pai son's tuibine, the steps in llu icsnltmg ex- 
pansion piocess aie so shoit that the pioccss becomes appioxi- 
mately continuous and may be rcpicsentcd by a continuous curve 
on the pressuic-A olume dia- 

giam 01 on othei diagiams 

A diagram such as fig 75 

then exhibits the complete 

action, the ontei cmvc BC' 

is a continuous line diawn 

thiongh points Mhichicpie- D 

sent the volume of the steam 

at the bcmnnmo of each slage, and the dilfeicnce bc'h\eui it .ind 

DO ~ ' 

the achdbatic cune BC shows how the \olumc is incuasid as a 
consequence of all the internal losses thai oeeui as the opualmn 
piocceds. The diagiam diffcis fiom fig 71 onl^ m tins, that tin 
cun c BC' is now to be undoisfood as including .ill inlciual 
fnctional losses instead of only nozvlc fur lion \\hal \\,is said m 
Ait. 110 applies to the efficiency of the luibuu ,is <i wlioh , and so 
long as no heat is lost by conduction the equation holds good, 

/-'-/.=-#- \\cnBCC". 

whcie Q is the heat gcncialed wilhm the tuibine b\ r fluid fuel ion, 
// is the total heat actually picseut in the steam il its exit I mm 
the tuibine, and /_, is, as bef'oie, the total heat \\Juch would be 
found in it aftci adiabatic c\|)ansiou to the same final picssme 

144. Theoretical Efficiency-Ratio. Whelhei the stages aic 
many 01 few, piovidccl no heal escapes to the outside by conduction 
or by leakage of steam, and provided the kinetic energy of the 
current of steam is negligible on its exit fiom the I in bine, the 
actual heat-drop 7 a / 2 ' is all icpiescntcd by woik done upon the 




v] JETS AND TURBINES 217 

rotoi. Let r} t stand for the ratio of the actual heat-drop to the 
adiabatic heat-drop. J$y this definition, 

7 7 ' 

7!,-- J 2 

^' ~ / _ 7 

2 1 *2 

Uiidci the conditions stated above this fi action expresses the 
cflicicncy-iatio of the turbine as a whole, namely the latio of the 
work done on the loLoi Lo the woik idc-ally obtainable by adiabatic 
expansion thiough the same langc The whole adiabatic heat-chop 
/! / 2 would be conveited into woik only if the tin bine weie 
revci bible and theicfoie thcimodynamically perfect. Owing to 
internal jrrcveisibility the heat conveited into work is less, apait 
from any lobb of heat by conduction to the outside 

We may call r) t the theoretical efficiency- 1 ) olio It is what the 
cfncicncy-i atio would be if the whole actual heat-chop I I I 2 ' 
wcie conveited into woik 

145 Action in Successive Stages The action of a com- 
pound tin bine is most, cloaih shown by using the Mollici diagiam 




FIR 7<> 

of cnhopy and Lolal heat lo exhibit what happens in each step. 
Beginning wilh Ihc initial pu-ssuic, let a senes of consLant-picssinc 
hues be diawn, p lt p 2 , p s , etc (fig. 7(i), concsponding to the 
piossincs at whicli tlic .steam cntcis the successive slaves In the 
lusl sUii>o the picssuic diops fiom p v to p z , m the second staoe 
fiom p lo pz, and so on In the fust stage, adiabatic expansion 
fiom p 1 lo pi would be ie])icsented bv a^, and the length of that 
line would be a mcasiuc of the adiabatic heat-diop, but the actual 
liCdL-dioj) is Ihc smallci quantity a^ Still ticaliug the action as 
adiathcjma], a^ is the heat converted into woik while the steam 
passes thiough the lust stage The condition of the steam at Ihc 



218 THERMODYNAMICS [en 

end of the fhst stage, and beginning of the second, is icpiesented 
by the point a z , which is found by drawing a line oi' constant 
total heat thiongh & x to meet the constant-pi essuie curve p i In 
the second stage adiabatic expansion would give the line r/,,r, 
The actual heat-chop, ^hich also measuics Lhe woik clone, is a z b i} 
and the condition of the s Learn as it passes on to the I had stage 
is icpresented by a A Snnilairy in the third stage the woik clone is 
3 &3, the steam passes to the fourth stage in the condition a. v , and 
so on The diagiam shows the pioce&s, as canicd down Lo I he 
boundaiy cuive, with steam initially supeiheatcd, it is readily 
extended into the wet legion In each stage the fiacLion fib/ac 
measures the latio of the woik done to the adiabalic heat-chop foi 
that stage The points 1} a,, a s , etc., he on what is called the 
"cune of condition," a cuive showing what the condition ol Ihc 
steam would be as it passes fiom stage to stage on Ihc assumption 
that no heat is lost to the outside The cui\e of condition conse- 
quently coricsponds to the outci cunc EC' of fig 75 The total 
\\oik done on ihe lotoi is the sum of the amounts of woik clone 
in the successive stages, namely 



146 Stage Efficiency and Reheat Factor. Taking any 
stage of a compound tuibine, the latio of Ihe woik clone to the 
adiabatic heat-chop, in that stage, may be called the stage eUicienev 
and denoted by 77.,, thus ^ 

7?s = ~ac ' 
The total woik done on the rotoi 



and if TI^ can be tieated as constant fiom stage to stage, 



The quantity 'Lac is called by some wuteis Ihe "cmuula,li\c heat- 
chop' " This quantity is gieatei than the \vhole adiabalic heal- 
chop between the initial and final picssmcs, /j - / 0) lo an cxlenl 
that depends upon the stage cflficiene}^ The uilio 

Sac 
/,-/, 

is called the Reheat Facto) The i cheat facloi is relatively high 
when the stage eHieicncy is low, 01, in other wouls, when Iheie is 
much Joss thiough irrcveisiblc action vuthm each stage 

* See Mr W J Goudio's book on Steam Tiubines (Loni;nians, 1917), p 1S)0 



r] JETS AND TURBINES 219 

Ti eating rj a as constant we have 

77, Sac woik done on rotoi 
^ s /] / 2 adiabatic heat-diop 

inclei the conditions postulated, which make the actual heat-drop 
L measure of the work done on the lotoi. 
From the equation _ ft 

t will he seen that in a compound turbine f] t is greater than the 
,tage efficiency 77,, since R is greatei than unity. 
We might have denned the i cheat factoi by lefeience to fig 75 as 

_ area ABC'D 

~ aiea ABCD ' 

bi in a compound turbine of many stages the cuive of condition 
s icprescntcd by EC' and the aiea ABC'D, which was called the 
'gioss appaient woik" in Ait 140, is the mechanical equivalent of 
.he ''cumulative hcat-diop" Tide. The woik done on the rotor is 
7 S x aiea ABC'D, and is less than the nicn ABC'D, the efficiency- 
ratio being ^ 77,_\_nica ABC'D 

f ~~ 



147 Real Efficiency-Ratio The foiegoing expicssions m- 
\olvc Ihe pioviso I hat llicie is no leakage of heal Hut when iheic 
is leakage of heal, or appicciablc kinetic cncig} in the steam at its 
e\il hum the luibme, Ihe actual hcat-diop 1^ // includes a 
quantity icpicsentuig the loss due to Ihcsc causes, in addition to 
Ihe woik done on the loloi Lei that loss be cxpiessed as a 
fi.iclion of the adiabalie heal-diop, naiuelv, 

'''(A-/,) 

Then 7, -4' -*(/!-/,) 

is that pail of the actual hcal-diop which is convcitcd into woik 
on the loloi. 

Hence allowing ioi this loss, the net or ical efdcicncy-iatio of 
Ihe tiul)ine becomes 



i r 

1 2 

since t] l is, by definition (Ait 144), the lalio of the actual heat-chop 
I] J 2 ' to the adiabatic heat-diop 

The amount of woik obtained fiom Ihc steam is therelore 



220 THERMODYNAMICS [en. 

Wilting t-j r foi the leal efficiency-ratio, its i elation to the other 
quantities is given by the equation 

In the process of designing a tuibme a value is estimated foi I he- 
stage efficiency y^, then the cuivc of condition is deduced, which 
allowsthe icheatfactoi to be found and also the piobablcvolumeand 
velocity of the steam at each stage In this way data aic obtained 
foi deteimmmg the toim of the steam passages. Details of the 
process will be found in books on the steam luibmc 1 . 

148 Types of Turbines. An "impulse" tmbinc is one in 
which the lotoi is cliivcn entiicly by the impulse of a ]d 01 ]cts 
against blades, which aic attached 1o it In such tin bines the 
expansion of the steam occms in fixed noz/Ics, or passages which 
act as nozzles. The tin bines of De La^al, Cinlis, and Zollv 01 
Ratean aic examples of the impulse type In DC Ii\ nl's I lie whole 
expansion takes place in one step, and the extinction ol encigy 
fiom the jets also takes place in one step Thcie is a single nng of 
blades, which must have an exfciemely high vclociU il il is to utilize 
a fairly laige fi action of the kinetic cneigy of the |ets DC 
Laval's tin bine is used only foi small powcis. Us efficiency is limited 
by the difficulty of making a wheel that will inn safclv at an 
enormous speed. The distinguishing chaiactciishr of the Cmlis 
tuibme is that the kinetic energy of the jets is exliaclod in steps, 
by making the jets impinge successively on two 01 nioic nngs of 
moving blades, with fixed guide-blades between to rlcflu-l llu ]ds, 
as aheady indicated in Ait. 142 This device allows of a mme 
pcifect conveision of the energy of the ]cls without icqunmg 
excessive speed on the pait of the i evolving blades. 

In some Curtis turbines the expansion takes place in a single 
stage Others aie compound in the thcimodynnnuc sense, the 
whole expansion is divided into a small numbci ol stages, and the 
kinetic cneigy acqmied m each stage is extracted by the use of 
a seiies of two or thiee nngs of mo^ ing blades In the Ratean 01 
Zolly type of turbine there aic many stages, each invoh ing a small 
chop of piessnre and consequently a moderate velocity of ]ct, the 
jets in each stage give up their eneigy by impinging on a single 
nng of moving blades Each ring lims m a scpaialc chambci , and 

* Sec also Batimann on "Recent Steam Tiuhino Piaoiico," Joiini fi^l Klccl 
Engineeis, vol ^.LVIII, May, 1912, 



JETS AND TURBINES 221 



he jets aie fonnod by nozzles or passages in the chap In agin 
eparates one chambei fiom the next 

A "leaction" turbine is one in which the* nozzles 01 passages in 
vhich the steam expands aie themselves the moving pait, and aie 
Iriven by the reaction which results fiom the fact that the steam 
s acquiring momentum as it passes thiough them An ancient 
oy descubed by Ileio of Alexandria, in which nozzles were caused 
o revolv e backwards by discharging steam into the air, is an 
xamplc of a pure reaction turbine. The type has not come into 
ise, it would rcqurrc an enormous speed of recoil to work efficiently. 
3nt a combination of icaction and impulse is applied in the most 
mpoitant turbine of all, that of Sir Charles Pardons, which was the 
list to be developed on economic lines, and is moie extensively 
ised than any other foi generating power on a large scale Parsons' 
s a compound tuibine with man}" stages. Each stage comprises a 
ing of fixed blades, projecting inwards from the case and making 
ip conveigent passages \vhich act as nozzles, and a ring of moving 
)lades projecting outwards from the rotoi The rings of fixed 
iladcs> and moving blades altei irate from end to end of the turbine 
aid aie alike m shape The mo\ ing blades, like the hvcd ones, 
nake up convergent passages \\hich die completely filled by the 
.Icani as it passes thiough In each set oi passages, moving as 
veil as lived, their is some expansion, consequently anv ling ol 
no\ ing blades is ingcd to move not onlv by the impulse ot the jets 
vlnch si like it, but by the reaction that arises fiom the expansion 
)t steam within it, since that expansion gives the steam new 
idocitv. The nencial chiection in which the sle.un ilo\\ r s thiough 
lie liubmc is paiallcl to the avis 

In an oail\ loim ol' I he Paisons luibinc the genual dncetion of 
low was laclial, the fixed blades being attached to a dvccl disc, and 
Lite moving blades 1o a paiallel disc which i evolved about ,m avis 
lluough lire centre of the fixed disc. An mlucsling modification 
>f this arrangement has been made by Ljungstiom, \\ho lets both 
"hscs revolve, but in opposite duecLions In the L|imgstioin tiu- 
binc (which is also compounded of many stages) theic aie, there- 
fore, no fixed blades, bolh sets arc urged by impulse as mil as by 
Leaction, and a high lelatne velocity, on which the stage elliciencj' 
depends, is obtained with a lower frequency of revolution 

Other turbines arc made up by combining the various types 
which have been named. 

It may be added that in compound turbines vvith many stages 



222 THERMODYNAMICS [en. 

the drop of pressme m each stage is so small that the nozzles., 01 
blade passages which act like nozzles, aie not of the con vci gent - 
diveigent kind described in Ait. 132. They aic only convergent, i'oi 
the drop of piessuie in each stage does not involve expansion beyond 
the "throat " In each stage the passages must be made sufficiently 
larger than those of the piecedmg stage to allow foi the mcicasc 
of volume that has taken place, in the final stages, when the 
piessme is approaching that of the condenser, the passages arc 
lelatively veiy laige. 

149. Performance of a Steam Turbine. In practice the 
steam tuibme, especially in large sizes with high initial piessme 
and high vacuum (that is to say, low pie^suie in Lhe condcnsei) is 
moic efficient than the piston engine, in the sense Ihal H conveits 
into effective woik a laiger fiaction of the heal which is supplied 
to it. For this icason, as well as foi its, gieatci mechanical sim- 
plicity, it has quickly come to be the duel means oi couveiLmg 
heat into woik on a laige scale, m pcwei -stations and in the 
propulsion of ships. 

As an example of its peifoimance the following figiucs ma), be 
quoted fiom a tual of a Paisons' tin bine emplo) cd to dine an 
electiic geneiatoi which developed about 5000 kilowatts The clcc- 
tncal output Mas measiu eel, along with the amount of t>kam \\lueh 
passed tin o ugh the tin bine in a gi\ en tune The luibinc iwis 
found to use 1319 Ibs of steam pei kilowal t-hom of cltctucal 
output One kilowatt-horn is the equnalent ot liWG thum.il units 
(pound-degiees ccntigiade) Hence of the heat-end g\ supplied 

in each Ib of steam r a or 113 7 theimal units wcie converted 
xo J. J 

into electucal eneigy 

If we allow for the loss of powci in the fncLion ol Ix-anngs and 
in the electucal geneiatoi, by taking the electucal output as 
94 pei cent, of the woik done on the lotoi, it follows lli.it each 

1137 

Ib of steam was doing woik on the lotor equivalent, to - - 01 

91 
152 9 theimal units 

The steam was supplied at an initial pressure of 21 ! 7 pounds per 
square inch (absolute) and was supei heated 07 to 2GI. 7 C The 
initial total heat was theiefoie 709 1 and the initial entropy was 
1 6257 

Thecondensei pressme was 47 pounds pei square inch Adiabatic 
expansion down to that pressure (under equilibrium conditions) 



JETS AND TURBINES 223 

mid piodncc a Avet mixtuie with a total heat of 4840 The 

mbalic heat-chop Avas theicfore 225 J- theimal units. 

Hence the uvtio oi' the work actually clone on the lotoi to the 

152*9 
mbatic heat-chop was ^ r or 68. This is the leal efficiency- 

Uo ^ ? If one might assume that the heat losses amounted to 
pci cent of the acliabatic heat-diop, so that x = 06, then the 
eorel icul elficicncy-iatio -r\ t Avould be 

f\t = i?i + 06 = 74. 

i'uiLlioi, the stage efficiency Aveie, say, 07 the coiiespondmg 
heal factoi would be about 1 06 These mmibeis aie conjec- 
lal, but tlicy may serve to ilhustiate the meaning of the seveial 
umlilies, and then geneial oiclei of magnitude 

150. Utilization of Low Pressure Steam As Avas buefly 
unUd oul iti Ait 95, the chief leason Avhy the steam tin bine is a 
cue efficient means of com citing heat into Avoik than the piston 
OUR, is its t>iculci power of making effective use of the energ\ 
low piessmc sLcam In the i eg ion of high pieusuie it has no 
\, ullage over the piston engine, but in the latei stages of ex- 
nsion il is a fai bcttei agent of conveihion, for it continues to be 
leient do\\n Lo Ihc hmcsl piesbiue that is piacticall\ attainaljle 
.) c'oiuknsei In a piston engine, on the othei hand, it uould be 
(kss ID cam expansion so fai, foi not onl} \\ould the bulk of 
( e\ limit i bceoiucimpiacticablc, but the mcicased \Ad^teof powei 
i oily li (uelion between the piston and the cUmclei \\ould become 
ealci I ban life gain ol indicated A\oik Hence, A\ith a piston 
gine, expansion in the cvlmdci is seldom in piactice cained beyond 
i absolute pies^me of 7 poundboi CA en 10 pounds pei scaiaie inch 
ilh a 1 1 11 bine I he expansion is continued ellecti\el) almost 
>wn ID Ihc condensei piessuie, and it is a mattei of the utmost 
nsc<nKiiee lo make that as low as the tempciatuit of the con- 
nsmg \v,ilc r will alloAs 

Tins poml \\ill be appmcnt if we use the entiopy-tempeiatuie 
agiam and coni])aic Ihc woik obtainable (undei ideal acliabatic 
millions) when expansion is complete down to a lou condenser 
essuu, Avilh Ihc woik obtainable \\hen iclease takes place at a 
cssmc'of say 10 pounds absolute In the diagram (fig 77) the aiea 
UCD i-cpiescnts the work obtainable m the complete acliabatic 
pansion of miUally satin ated steam fiom a piessuie of 130 pounds 
a condensei prcssiue of 5 pound, and the aiea ABCEF repre- 
nls Ihc work obtainable when release takes place aftei expansion 



THERMODYNAMICS 



[err 



do\\n to 10 pounds absolute, EF being a line of constant volume 
(Ait 96) The same condenser piessuie is assumed in both cases 
The aiea FED icpiescnts what is lost by incomplete expansion, 
such as necessarily occuis in a piston engine The figure applies 
to an ideal peifoimance in each ease, with adiabatic expansion, 
but in the conditions of actual woik the steam turbine would save 
most of the aiea FED. It is to be noticed thai any i eduction of 
vacuum will dimmish the output of work J'iom Liu Liu bine much 
moie than it will dimmish the ouLpuL horn the piston engine, 
foi when the line AD is raised it afteets the tin bine aiea ^ 




2 



along the whole length of ,1/Vwhcieas U alfecls the pMon-uM>me 
aiea ASCEF only along the short distance IF 

The importance ot high \acimm in a slcam Imbim is besl ivd- 
ized by woikmg out values ot the adiabat.c hcal-d.oj, w,(h A ...nms 
back pressuies. Taking initial conditions such ,,s ,,.i found u, ,, 
tice, with p ieS Muie anywheie between, say, 100 and 200 pounds 
pei squaie inch, and modeiate supe.heal, the mlil>nl.c hc,,(-dm,> 
is mcieased about 10 pei cent uhcn the eonden.se, nussmc , s 
leduced liom 98 to 19 pound, which couesponds lo ., ,,n- 
piovement of vacuum liom 29 Lo as mchts, w.th the kuometer 
at 30 inches 



CHAPTER VI 

INTERNAL-COMBUSTION ENGINES 

151 Internal Combustion. In an internal-combustion engine 

the fuel which is to supply hcat-eneigy for conveision into work 

forms pait of the working substance, and its combustion takes 

place within the vessel or system of vessels in which the woikmg 

substance does woik by expanding. The woikmg substance, theie- 

foic, undeigoes a chemical change cluung its opeiation and the 

bhcimodynamic process is not cyclic In the eaily stages, before 

combustion, the substance is a mixture of fuel with an, geneially 

in excess of vdiat is icquued to prov ide enough 0x3 gen toi complete 

combustion In Lhc Litci stages, aftci combustion, it is a mixtuie 

>f the pioducts of combustion with, mtiogcn and svith any suiplus 

>1 air The lucl commonly enteis as a gas 01 v apom diawn in along 

iViLli a MII table piopoi i ion of air, but it may be injected as a liquid, 

jecommg vapon/cd aflci admission or ducctly buint on entiy 

\s a mlc the only chemically active constituents of the futl aie 

i)du>gui, h vdioc'tiibons and cai borne oxide In then combustion 

hey mute with oxygen to foim watci-vapoin and caibomc acid 

riu 1 nitiogcii of Ihe an takes no po.it in the chemical pieces-. 

K-yond acling as a diluent 

Typical examples of internal-combustion engines aie the oidmary 
jas-engme 01 the pcliol motoi, in which a "chaige" of an mixed 
vith combustible gas 01 vapon/ed liquid fuel is drawn in by the 
)iston, then composed into a cleaiance space, and theie ignited 
>y an electiic spaik or other means, so that explosive combustion 
akcs place while the volume of the chaige is neatly constant. 
L'he heat I hi is internally developed gives the working substance 
high tempeiature and piessure it then expands, doing woik as 
he piston advances. In all modem engines of this class the charge 
bi ought to a fairly high pressure before being ignited. It will 
>e shown later that this compression secmcs thermal efficiency, 
pith inci eased compression a larger fraction of the heat of com- 
ustion of the fuel is converted into effective work. 



226 THERMODYNAMICS |rn 

From the theimodynamic point of view mlernal-comlmslion 
engines have this advantage ovci the steam-engine, lha.1 I ben 
woi king substance '"lakes in" heal (bv iK <>un combnslion) al .1 
much Jiighei tcmpeialme In the (oiHl)iislioii of I he chmge n lem- 
peiatuic of 2000 C 01 so is leaelied The avciagc lempeial me ,il 
which the heat is developed is far abo\ c thai al which lu'al is 
icceived by the working subslanee of a sleam-cngine On the- olhci 
hand iL is not piacticable Lo diseluige heal al neailv so low .1 
lowei limil But the aelual working lange ol Icmpeialuit' is so 
wide that a gas-cnginc can in f.iel eonvei I mlo woik a laigei 
fiaction of the heat-cncigy of the Inel llian is eonvei led !>N an\ 
engme which bums its fuel lo laise sleani in .1 boili r, and uses I In- 
steam, however efficiently, as woi king subslanee \. good gas-engine 
will conveit about 30 pci ccnl of I he eneigy of ils IIK 1 into uoik 
the best steam-cngmcs eonvei I no moie llian aboul L() pc-i e nl 

152 The Four-Stroke Cycle Tn I he most usual lype <>l in- 
ternal-combustion engine the mcehinieal < yele is eomph led MI loin 
stiokes 01 two revolutions Dining I lie In si loi w.nd slmke, gas and 
air are diawn in, so lhat the whole e> lindei is lill< d \M! h < \|losi\e 
mixtme, at practically atmosphci ie piessuie Dining llu Insl l.uk- 
stroke this mixfcme is compiessed mlo .1 eh aianee space al I IK ( nd 
of the cylmdei The mixluie is Ihen igmUd, \vlnl( llu pislon is ,il 
or close to the "dead-poml " 01 exheme ol ils ha\ 1 1 The pn SSIIK 
consequently uses to a much lughei \alne llian w.is icaclird li\- 
compression. Dining (he second forw.nd slmke Hie liud nu\lui< 
expands, doing woik and falling in pussme Dining UK s<<oiid 
backstioke it is dischaiged lluough ,m e\li.iiisl-\ al\ e nilo UK 
atmospheie A small quanlilv of llu bninl nuxlini K mains m 
the cleaiance space, and is nu\ed wilh Ihe IK \l (hai'u unless 
special means aic taken lo lemove il, by whal is e.dNd tl sca\( n- 
ging." As a rule theie is no scavenging 

The fom-stioke cycle was ihsl deseubed by lienu de Itoelias in 
1862, it was biought mlo use by Olto in IS7(>, and is ollen called 
by his name. IL is si ill the most usual mode of achon, nol with- 
standing the practical diawback of having only one woi king si ioke 
out of fom, a drawback which arises liom Ihe fael Ihal Ihe working 
cylmdei seives also as inhaler and compiessing pinup 

153 The Clerk or Two-Stroke Cycle To escape this d< led 
of the Otto cycle, Sir Dugald Cleik mtiodueed m 1H81 mi c'ngme 
wliich completes its action m Uvo strokes Clerk's Engine has a 



228 



THERMODYNAMICS 



LCII. 



in the (iist stroke ol the next cycle. It' the ideal engine were of the 
two-stioke cycle type, the lines AB and BA would be omitted from 
the indicator diagiam for the woiking cylmdci, which would Lhen 
consist simply of the figuie BCDEB. 

From C to D the whole heat-encigy developed by the combustion 
ol the chaige goes to heat the woiking substance, since by hypothe- 
sis none is lost by conduction or lachation to I he walls. The heat 
of combustion can be calculated when the composition of the chaige 
is known, 01 may be measmed diiectly by binning a sample of the 
gas in a calonmctei In all cases one of the pioducts of the com- 
bustion is watei -vapour, and as any water-vapour foimcd m the 
cyhndei of an internal-combustion engine remains uncondensed 




Fig 7S 

Llnoughoul the action it is pioper to lake, in calculating Hit lual 
dc\ eloped b} combustion, what us called Ihe "lowci " value, that 
is to say, a value which docs not include the lalenl heal of Ihe 
\vatci-vapoiu 

Between C and D the mixture undcigoes <i chemical change 
which may 01 may not affect its specific volume I hat is to say, Ihe 
bin nt products when bi ought to the same piessuie and tempcraluic 
as the un burnt mixture may not fill exactly Ihe same volume In 
gcueial the specific volume after combustion is a little less, but 
with such mix tin es as aie used in gas-engines or petrol-engines the 
effect of this "chemical contraction," as it is called, on the specific 
volume is so small as to be ummpoitant With mix tin cs of coal-gas 
and air it amounts to between two and three per cent, in ordinary 



vi] INTERNAL-COMBUSTION ENGINES 229 

cases With some explosive vapours the specific volume is slightly 
increased (see Art. 158). The changes being in any case small, it is 
convenient in considering an ideal engine to ignore them, and to 
treat the working 1 subs lance as if it weie a gas whose specific 
volume docs not alter Fmthei, the largest constituent of the un- 
burnt charge is an, and that of the burnt charge is nitrogen, and the 
specific heat of nitrogen is,foi equal volumes, the same as that of air. 
Hence for the purpose of. obtaining a simple standaid with which 
ical engines may be compared, a practice has sometimes been 
adopted of tieatmg the woiking substance as if it wcie air., to which 
between C and D theie is unpaited a definite quantitjr of heat, 
namely the heat of combustion of the chaige. 

155 Air Standard It was on this basis that a Committee of 
the Institution of Cuil Engmeeis* devised what is known as the 
"An Standaid" as a meaMiic of the ideal efficiency of an internal- 
combustion engine 

Resides assuming (as in Ait. 15-1) foi the puiposcs ol then ideal 
slandaid 

(1) No tiansfei of heat between the woiking substance and 
Lhc metal, 

(2) Instantaneous complete combustion, 
and (.'3) No change of specific \ohune, 

I hey made I he fm Lhci assumption 

( I) That Lhc specific he.it might be healed as constant (inde- 
pendent of tcmpcialme as ucll us pic-ssme) 

II is nun u'cogm/cd that Lint, last assumption is by no means 
line c\cn nl an, and is slill moic unLiiic oi the mixed gases in I he 
cylinder ol a gas-engine- It is known that the specific heat mcicases 
vuth use oL Icmpciatuie to an extent which gicatly affects the 
action of the engine This point will be considered lalei but it 
should be said line that because the specific heal of Lhc woiking gas 
is much oic'ulei al high (cmpcraluics than at low tcmpcialiues, the 
u au slandaul," as defined by Ihc Commitlee, is an umeasonably 
high ciitciion Lo apply to any actual perfoimance. The efficiency 
of a icjil engine must fall ic<illy short of lhat standaid, not only 

* Iti'jHnt, of a (Jommillot) cm the Ellicionoy of LntcinaL-Comlmslion Engines, 
Mm Pioc Just C E vok 1 02 amU03 (1905 and 1900) Tho Rcpoi I gives examples 
ol oiilculaUoiiH i dating Lo actual and ideal poiioiraancos Kefoionce aliould also be 
jiuido m this connection lo SH Dngald (Jleik's book on The, Qti^, Pcliol and Oil 
JfniM, veil T 



230 THERMODYNAMICS |cn. 

because of such moie or less avoidable losses as occur through 
radiation and conduction of heat to the cvhndei walls, bul because 
the standaid postulates, on Ihe pail of the woiking substance, fin 
essential quality that is widely diffei en t Irom (he quality ol thcieiil 
gases of which it is composed JKxcn if Iheie were no loss of heat, 
the limit of tempeiatmc which the gases reach aflei explosion 
must be much lowei than that which would be reached if 111* 1 
specific heat weic constant. Ilowevei much Ihe heal losses <iu i 
minimized, the hypothesis of constant specific heal makes the air 
standaid an impossible ideal 

It is nevertheless instructive to study the an standard with 
constant specific heat as a means of examining some ot the ellecls 
that follow fiora vaiying the conditions of 
woiking We may apply it foi instance 
to show hoA\ the efficiency ol the gas- 
engine cycle is impioved by increasing the 
compiession 

Let T and T^ be the absolute tcmpcia- 
tures of the charge befoie and aftei com- 
pression, and let T" 2 and T s be the Icmpcia.- 
tuies befoie and after expansion Fig 7<) 
shows the cycle with its stages numbeied 
to coriespond with these si ilfixcs Wnlc/ 
for the ratio in which the charge is com- 
pressed befoie ignition, which is also (he ulio in \\lnch il is 
afteiwaids expanded dm nig ils woiking shoke Then by Ail '_'<>, 
since the compiession and expansion aie assumed lo be adiabalic, 

-. CT S ' r ' 

fiom which also Tj ~ 7 " -= r <> ( 1 } v ' 
^i ~ TI '/', V? ' 

Hcie y is the ratio of K ;| Ihe specilic heal al conslanl piessme 
to K a the specific heat at conslanl \ ohmie and is I realcd as a con- 
stant because the specific lic.ils aie assumed lo be conslanl m Ihe 
"an c^cle" whose efficiency we aie now finding 

The heat supplied, namely the heal flcnoiMlnf in I he explosion ,s 
K (T, - T,) The heat rejected is K,, (T, 7 1 ,,), for il makes iio 
chffei encewhethei the products of comlmslioiune cooled on release 
to the atmosphere, or kept in the cvhnde. and cooled there to 
atmosphenc temperaluic.aL conslanl volume, before being released 



vi] INTERNAL-COMBUSTION ENGINES 231 

Hence the thermal equivalent of the woik done in the air 
cycle is j IT T } K (T T \ 

and the "air-standard" efficiency is 

K r (T 2 - T : }~K r (T. - ?') 1\ - T 

K a (T,-'J\) m } ~ f'^TS 

/I \ yl 

which is equal to ~ 



This c'xinession is nnpoil.inl as showino the beneficial influence 
of compiession. Much of llie pracLical iinprovcmenL of oas-cnQiaes 
has in fad lesultcd 1'ioni pio<)icssi\ r cly mcKMsuiq Ihc cxlent to 
which the chaie is coniiJiessed bel'oie i^inlion 

WjLh incieascd lalios of compicssion the "an-sLandaid" effi- 
ciency ineuases as follows, lakinn y to be 1 I. 

Jlalio of Vii-StniuLinl 

n lillic ii'iK y 

2 U212 

{ !, r )() 

I I) 12(} 

r> o 1.7^ 

( Of) 11 

7 Of) II 

S I) r )() r ) 

10 OM)J 



II will be seen lioiu Ihcsc II^IIKS .ind lioiu llu' cui\c (|IL> ( SO) 
I luil lluie is al liisL a \ ei v i.ipid ^><IIH ol elluieneN \\ill) mcie.ised 
compiission, buL IhaL ^\llLu UK compulsion is lu^li Ihe llieimo- 
d} namie advanLi^e of iiH-icasiiio il boconu^ sli^hl \Vheu accounL 
is Likeu of vaii.ilion m specific heal, liyuies aie obtained foi the 
Lheort Ueal (.Ilicunev which fall slioil of Lhe an sL.mdaid by about 
20 pei eenl (see Ail 107), buL Ihc proporhon between Ihe elli- 
jiencies for dd'fercnl amounts of compiession is nol ^>ieaLly alLeicd. 

The ellieiencies aelually obLmicd m trials of engines are of comse 
considerably lower, owing to heal losses and to the fact UuiL tlie 
combustion of the chaise is not instantaneous. In veiy favourable 
cases the measured thcmial efficiency is as hiyh as 37, coircspon- 
-hng Lo about 08 per cent, of the an staudaid, 01 to about 83 per 
jent of the theoretical standard that is obtained when account is 
.iikcn of variations m specific heat This is for engines of the 



232 



THERMODYNAMICS 



[C'H 



oidmary tj'pe in which combustion occurs, al nppioxunalcly con- 
stant volume, aftei the compression ol a mixed charge 

In all such engines theie is a pi action I luml lo I lie amount of 
compiession' it must not be so great as to cause prc-iimilioii by 
unduly laismg the tempeiatme befoie the end ol the compression 
stroke This limit diffeis with different kinds of 1'uel, it is com- 
paiatively low when theie is much hydrogen. In engines using 
ordinal y coal-gas the latio of compression is in piachec as hii>li <is 
6 01 7, m petrol-engines it is usually about '1 01 less We shall sec m 
the next aiticle that by departing from Lhc constant -volume I yju* ol 
combustion, higher ratios of compiession become praeheablc, willi 
some mciease m theoretical thermal efficiency 




'20 



u 2 4 6 8 10 12 14 

Ratio of compicssion. r 
Fig 80 Eflicioncy of " air Simula id " 

156. Constant-Pressure Type Besides Ihe consUnl-N chn.u 
type of inteinal-combustion engine, to wlueh oidiniiiy g;is-c nu m < s 
and most oil and petiol-engmes n])piox,nui[cly conlin.n, \\^ni,n 
imagine a type in which the piessuu- of Ihe workinu subslancc 
does not change while combustion is lakmg place Si.p,, ( ,s ( (h-,1 
the an is sepaiately compiessed mlo the cle.uuncc- space belo, e a 1 1 v 
fuel is admitted and that fuel is then fenced m, bununo- as ,1 nuV, s 
while the piston begins its foiwaid movemenl Jiy suKably ,e<M,- 
latmg the late of admission of the fuel Ihe ]),essu,e may be kri,l 
constant till the combustion is completed. 

In this imaginary cycle the heat is supplied at eonslant p.cssmv 



vr 



INTERNAL-COMBUSTION ENGINES 



233 



We may furihci imagine the icjection of heat to occur at constant 
piessuie, if we suppose that bhe products of combustion aie ex- 
panded adiabatically do^n to almosphciic pressme befoie they aie 
discharged. The ideal indicator diagiam would then take the form 
sketched in fig HI. Undei these conditions (which are not leahzed 
in practice) AVG should have an engine of constant-pi essme type, 
i ejecting as well as receiving heal at constant piessure Its au- 
jtandaid efficiency is icadily cxpiessed in a foim coi responding 
o IhaL found for an engine of constant-volume type We aie 
once mod heic with Hie specific heat at constant pressme, K v 




F:^ SI Coii-il mt purlin. 1 h pi 1 

'icaling it MS consUml, the heat taken m is K p (T, T t ), the 
eat rqeclecl is K (T A - T ), and the efliciencv is 

A- / rit //' \ /' 

/ ( 7 2 J - J l) ~ K I> 



'he lalio oCadiabahc expansion is equal to I he latio ? of adiabatic 
nuiwession, 2\,/7\ = T/l\ Hence the aii-standciid efficiency is 
iven by the same expies.sion as befoic' 1 , namdv 

1 - (T 

w 

* It is mleiosling to noto tliat tins same oxpiession applies to three ideal types of 



igmo 



(1) Tho constant-volume typo, m which Iioat is ifceived and rejected only at 
constant volume 



234 THERMODYNAMICS fc-ir 

It follows that foi equal ratios of compression thtvre would ho no 
thermodynamic advantage in substituting a constant -pressure 
type of engine for the constant -volume tvpo Hul hy avoiding any 
admixtuie of the fuel with the an before eonipiession il heroines 
piacticable to use a higher ratio of conipiossion, and consequently 
to obtain a highei efficiency 

157 Diesel Engine This advantage, is in part secured in the 
Diesel Engine., which compresses the air sepaiatelv (o a |)iossiiro 
of 500 pounds per squaie inch or nioie, bc/oic (he I'uel is admitted 
The an is compressed by the baclcwaid slioke of (he pislon Tin- 
fuel is oil, which is delivered by a separate pump into I lie highly 
compiessed an while the piston begins its foiward slioke The oil 
at once ignites, because of the high temporal me lo which I he ;m 
has been biought by compulsion Us combustion keeps I IK pies- 
suieneaily constant until the supply of oil is cut oil 1 Tin pioducls 
then expand, but not to the extent shown in tin iin,igin,n\ engine 
of fig 81, for expansion is continued only to u \olnme |iul lo th.il 
of the an befoie compiession, so that when lelo.ise Likes place, 
the piessuie is much highei than that ol'lhe .ilmospheic As leg.uds 
leeeption of heat the action of the Dnsil engine .ippio\ini,iles (o 
the constant-pi essine type, hut as ugjnds lojcchon ol lu.il i[ 
approximates to the constant-volume type, and in (hat i< sp I i( s 
theoietical efficiency is somewhal uduced The high iiiih.il eom- 
piessionenahlcs it, howcvci, toeonveil moicol I he I luiui.il em IH\ 
of the fuel into indicdled woik than is eon\ oiled in olhei mh-iii.Tl- 
combustion engines The piaelieal ad\.inl.ig< ( ,f || lls ls j,, S(MU( 
extent countei balanced hy its gi-e.il ei mcch.uuc.il lueli m , ulm-h 
brings the net output of effective woik down toaliguie moie n()1I | N 
compaiable with that of olhei engines 

In a tnal ot a Diesel cngmt indicating nenilv KM) huiM'-pmu i l! 
a theimal efficiency (with legaid (, mdicalcd poue,) ,,| o 17 , s 
claimed, with a mechanical elhe.enoy ol () 7<> II ||,,-si. lig,,,, s .,ie 
conect the engine was eonvei tmg :j<j pe, cent . ol I he I he, nul euei ,r v 
of the fuel into effcetn c woi-k, available for d, u ,, ol he, machin^ 

(2) The constant piowuio L yjlU) m w |,u h lu, ll( , , ( ,uv,,l and j, , (l ,1 ..... v llL 

consLaiit picasiuo 

(3) Tho constant LompouituuUy^ {('aiii|,' H .. MK n. nf li { r I, A.I ,I(,J in uhi< h 

heat is tooivod^uul .ojoUod only al, (i..n H liuil lnm|>i-iu(iu<. Kui Kn 
efflcianoy I - , a ,l ' 1IH1 , , , (l( , , ||U||1 (|f 



adiabalu. compression (nut isotliouual, an in Ai 1, 
Quoted by Mr Mathol, / OM , l )M n fllll K ll .\h 



vil INTERNAL-COMBUSTION ENGINES 235 

158. Combustion of Gases. Molecular Weights and Volumes 
In calculations that iclntc to the combustion of gases the 
quantities involved aie most conveniently reckoned per unit of 
volume, at a slanchiid condition as to tcmpciatine and pressuic 
A chief reason foi this is that the densities of oases are piopoi tional 
to then combining weights, and consequently the volumes m \vhich 
I hey nmle have a very simple ratio. 

The combining weights Poi the substances with which we aie now 
concerned aie (in lound nuinbeis "), 

Ilydiogcn II = 1 

Oxygen 010 

Nitiogen N = ]-J- 

Cnibon C =12 

Ilydiogen, ox) r gen and nitiogen arc diatomic gases, that ib to 
say their molecules, II,,, 2 , N,, each eompnsc two aloins, and then 
molcculm \\ughK aie accoidmgly 2, 32 and 28 icspecUvcly The 
volumes represented by these weights aie the same for all Llncc, 
when biought to the same picssiiic and teni[)ei<itine 

The equa tion 21 1, + O , - 2 1 F ,O 

means that in the combustion ol hydiogen two molecules of hydio- 
gen unite Milh one molecule of oxygen to fonn I wo molecules ol 
watei As legaids weights, it menus lhal I pails by \\eigl\l ol 
h>diogcn unite with 32 pai Is bv weigh! ol o\\ gc n to foim 30 |>ai Is 
by weighl of walci, and lhal I he moleculai \\ciylil of water is IS 
As legal els \olumes, it means lhal I wo xoliiuus <>l h\diogen unilc 
wilh one \olume ol o\vgtu lo foim two \olinnes of walei-\i[)om, 
assuming lhat the compai isou of \ olunus is made nuclei such con- 
ditions of U mpcialme and piessme that I he waUi-vapoui may be 
ticalecl as pei feel ly gaseous 

Again, the cqualion 2CO -h O, = aC'O., 

means lhal two molecules ol cai borne oxide (SO pails by weight) 
mule wilh one molecule ol o\ygc n (32 pai Is bv w< ighl) to Joi m Iwo 
molecules of eaibonie 1 acid (SH )>ails by weighl) II also means that 
two volumes of eaibonie oxide mule wilh one volume ol oxygen 
lo lonn Iwo volumes of eaibonie ticid. 

In Ihe combination of any gases Hie pioj)oi tion by volume is 
given clnoetly by I he iclative munbei of molecules The pimciplc 
involved known as Avogaeho's Law is lhat equal volumes of all 

* More exactly, taking as 16, II is 1 008 and N is H 01 



23(J 



THERMODYNAMICS 



[en 



gases (in tlie peifcctly gaseous state and undei the same condition*- 
as to piessure and tempeiatme) contain the same uumbci of mole 
eules The weight contained in unit volume- in olhci words the 
density is theiefoie proporLional to the moleculai weight 
A feu further illustiations may be useful 



Weights 
Volumes 



Weights 
Volumes 



Weights 
Volumes 



Weights 
Volumes 



Marsh gas (Methane) CIIj 

CH 4 + 20 2 = C0 2 + 211,0 

16 + 01 = 1-1 + 36 
1 + 2 foim 1 + 2 



2II 2 O 



ELhvIenc CJIj 
C,Hj+ 30 2 = 2CO> 

28 + 96 - SS -I- 36 
1 + 3 form 2 + 2 

Butylene C 4 II 8 

c 4 H a + 60, = jco 2 + 4n 2 o 

56 + 192 = 176 -|- 72 
146 foim 1 4- -J< 

Alcohol C,II 0. 

C JI^O H- 30, == 2C0 2 + 311,0 

JO + 96 = 88 H- 51 

1 + 3 form 2 + 3 



It will be obseived thai with alcohol and \\ilh llu hcav} hydio- 
caibons, of which C 4 II S is one, the specific \olnmc is mei cased by 
combustion, wiLh maish gas and clhylcnc il undeigoes no change , 
and Avith h^ydiogen and caibouie oxide iL is i educed The clhinne 
of specific volume which any given gas nnxLuie will muh'igo on 
complete combustion is leachlv piechcLcd by a]^|)l} ing I Ins method 
of calculation to each of Lhe constituents oJ the fuel, when Ihc 
composition of the mixtuie is kuoun AnoLhcr obvious ji|)[)hcalion 
is to calculate the volume of oxygen, and by mfeicnce the volume 
of an, requned foi the complete combustion ol a given gaseous fuel 
For the puipose of such calculations, cliy air may be laken as a 
mixture of 20 9 pei cent, by \ olnmc of oxygen wilh 79 1 pci ei'iil 
by volume of mtiogen 

The folloAvmg example will seivc to show how I he an required 
for the complete combustion of a gas of known composition js 
calculated, and also the change of specific volume, 01 the "chemical 



[J INTERNAL-COMBUSTION ENGINES 237 

mtiaclton," winch will take place on combustion The fuel is 
>al-gas 5 of the composition shoAvn in the fiist column. 



Vol ot oxygen 
Composition of lequiredfor 
the gas < ompleto Volume of pioducfcs 
ly volume combustion H 2 CO, N, 
H 2 422 211 422 


('[! 


3; o 


680 


08 34 





G n H 4 


48 


144 


96 96 





CIH 8 
CIO 


2 1 
(i4 


126 
32 


84 84 





JSL 


08 








OS 


c'Oj 


37 





37 







1000 


1193 


128 2 62 1 


08 






210~3 


Hvfl 





ic chemical contiaction is 219 3 197 1 = 22 2 volumes. 

Since 119 ,'3 volumes of oxygen aie leqiuiecl foi complete 

i , M i 110;J " 10 e ~i i 

mbuslion, I lie an lequued is t 01 571 volumes. 

wW J 

I'ncc il llns gas is c \ploclccl in the nchcst possible mivtine, with 

smpliis an noi olhoi chliicnl, the contiaction amounts to 22 2 

a lolal \olnmc ot 071, 01 33 pci cent In a gas-engine \vheie 

( mixluit 1 is diluted bv excess an, and bv lesidual pioducts 

>ni a pic'Mons cliaiqe, the contiaction \\ill of coin se be a smallei 

ipculion of the whole volume 

159 The Gramme-Molecule or Mol The essence of Avoyaclio's 
[iifiplc may be- put in anolliei \vay by sayni" Lliat if we take 
mills (i>i, mimes 01 I!)b ) of diffoicnl oases, ni bcmy the numbei 
ucli K'picsents the mokcnlai weight, I heir volumes (at standard 
jssiiu 1 and leinpeiriLuic) will be Lhc ^ame- 1 Say, foi example, 
il Liu 1 chosen unit of mass is Lhc giamme take 32 giammes of 
ygen, 28 gianuncs of mLiogen, 2 giammes of hychogen and so on, 
j volume oi each quantity will be the same Ic This quantity, which 
hi lei on I foi diffeient gases, but has neaily the same volume foi 
, is called a "giamnic-moleculc" ot "mol " iLs volume is 22,^00 
)ie ecntnncLres for Lhe nearly perfect gases, at C. and one atmo- 
icie f . Pioj)eitics of gases such as the specific heats, or the internal 
, or the heat ol combustion, aie often stated per "mol." 



Subject to small differences which aie due to the fact that the gases are not 
jtly "perfect " 
Calculated from the density of oxygen 



238 THERMODYNAMICS [en- 

A quantity of heat stated m giamme-caloiies per mol may be con- 
verted into foot-pounds pei cubic foot by multiplying by 



002205 x 1-100 

or 3 90, 



(0 082808 px 22,-! 00 

since 1 gi.-caloiy = 002205 Ib -caloiy, 1 Ib.-caloiy = 1 MM) foot- 
pounds, and 1 cm. = 032808 ft. 

160. The Universal Gas-Constant The i>,is equal ion 

PV = RT, 

is strictly applicable only to ideal oases winch aic ''pi il'ecL" in I ho 
sense of oiling Boyle's Law and also Joule's Law (Ail 10), 7' 
beino the absolute tempeiatuie on the theimodynanne scale It is 
appioximately tiue of all gases at low or moderate press mes, pio- 
vided the conditions as to piessuie and tcmpciahuc arc nol such 
that the gas approaches liquefaction. At any given leinpeialiiu- 
a leal gas is moie and moie neaily "peifecL" the more the picssiiu 
is icduced. Wilting the equation in the fonn 

RT 

V 

p ' 

and multiplying both sides by m, the nunibci which cxpiessts I he 
molecular weight, we have 

rr mKT 
mV= p 

Heie mV is the volume of ? units ol mass Bui thai volume, as Mi- 
saw in Ait 159, is veiy nearly the same foi all gases nuclei I lie same 
conditions of piessuie and tempciahnc Hence in/i is also \eiy 
neaily the same foi all gases The qnanlily in It is called (lit nm- 
veisal gas-constant Like R (Ail 18) it is a qnanlily ol \voiL, lobe 
expiessed in work units 01 equivalent heal umls Us numeiieal 
\alue depends on the unit of mass that is used m the leelvoning. 
Let the unit of mass be thegiamme, IhcnwFrcpifsuils I he volume 
of a giamme-molecule or mol, which is 22,400 cubic centnnehes 
when the teinperatuie is 0C. and the piessuie is one atmos|)heu 
(equal, by Ait. 12, to 1032 7 giammes per sq cm , or, m absolute 
measure to 1 0133 x 10 dynes per sq cm ) 
Hence the universal gas-constant 



D 1032 7 x 22,400 

= 273 1 - = 8I '' 700 



-ij INTERNAL-COMBUSTION ENGINES 239 

-Ve mav also express the gas-constant in hcaL units Since, by Ait. 
4, the giamme-calory is equivalent Lo -126 7 giamme-mctres 01 
'2,670 giammc-cenlimetrcs, 

ni If - l =1 085 giammc-caloiies* 

12, (5/0 

Igam, il Ihe mill of mass bo I ho 11) , Lhe "as-constant cxpiesscd in 
b-caloncs is 1 085 | s which is (qui\alcn(- to ] 985 x 1400 = 2779 
DO I -pounds 

The gas-conslani may be inlcjpielecl as Ihc woik lhat is done 
y expansion when in unils of a gas aic healed nuclei oonsLant 
lessmc lluough one degree 

Knowing the ga.s-consla.nl we can icmlil}' caloulaLc Ihe value of 
J 111 the equation PT -- KT (01 an\ gas Lo which llial equation 
pplics, bv di\ ulmg Ihe constant b> in Values of If calculated in 
us mannei aie gi\ui below 

('iili'iiliilf (I valuos df Jf 

111 <ri UIIIIK I Illlllli i |)l I <r|,lim>l(i 111 llHll |)(IHlllK 
HI II) ( ll(>l II 1 |)l I 111 [)l I II) 

(Kvgta ()()_>( I 8(>S 

Nido^LMi 0070') )') J 

Au (MKiSS ')(. { 



Cuboim <)\itlc 0070') <)!> ,] 

II should be icealltd lh.il Ihe \alue of If is equal lo Iht dillti- 
ice Ix'hvcon Ihe specific lu.ils ,il conslanl pitssme and ,il eon- 
anl \olnme, I\ and A',, (Ait. 20) 

161 Specific Heats of Gases in Relation to their Mole- 
ilar Weights Volumetric Specific Heats In Ail 20 we 

11 UHIIHJ iiliMoluit (r <r H ) uiuls, (he slanclfinl iilmosphuiH as doJmod in Ail U in 

H'$.J > JO" d\ IK H J)( 1 H([ (111 JIlllUO UlO gllS COIlHltinl 

1 01. 'W % 10" % 22,400 

21, \ J. ~ L ^ 

o guiinmo-caloiy UH dofinod in Ail 10 is oi^uivalonl to 4 1808 A 10 7 oigH honco 

S3 n , 10" 

? " A>= -4]H(>HxU,^ ]08 ^ UttmmOCalOU08 ' 
in tho text 

Tho numoi ical value la uol allot od, 1 Ib calory por Ib being equal lo I gramme- 
ory por giammo 

Taking m for Iho mixluro of nikogon and oxj^gon as 28 86, namely 
79_1 x_28 20 9 x 32 
100 "' " 100 



240 THERMODYNAMICS [<'"- 

reckoned the specific heats K p and K per utul of mass Foi many 
pmposes it is moie convenient to icckon the specific heats of jjasi-s 
per unit of volume when so icckoned they aie sometimes called 
volumetric specific heats Most eoiivemenl of all is lo icekoii I hem 
per gramme-molecule or mol. This ism olTeeUi volumelne method, 
foi the volume of the mol is Lhc same m all t>as( s lh.il salisl'v tin 
equation PV - RT When the specific' heals of such gases are 
reckoned pei mol their diffeience, is C([iial to the gas-constant. 

TIlUS K, = K v -[ 1 1)85, 

when K v and K v aic reckoned in gramme-calories per mol 
It follows that in all such gases the mho y of A",, lo A',, is 

1 OS 5 
y = I+ K, 

The vohimetiic method of icckonmi> specific h< il h.is ( Ins hn lliei 
advantage that when so reckoned I he spirilu lu.il (A' ; , 01 A,) 
of the -simplei ga^es is neaiJ^ the same, [iiovidcd the ij.iscs h,i\< 
the same nuinbei of atoms in the molecule Ml the y.isis ii.nncd 
m the list in Ait 1GO aie diatomic, .null he\ h.ncm.iiU Hit s.nuc 
specific heat ^vhen that is leckoucd [>ci unit ol volnnu , 01 p< i mnl 
This is found to be tiue when the spc'cilu heals of lh<-s< g.isc s an 
expemnentally measiucd and comp.ucd, bill il c.ui also hi- inl< IK<! 
fiom the kinetic thcoiy of gases (See A|)[)i i ndi\ II ) 

The kinetic theoiy shows thai m an ulial 1 diatomic ^as 
Ky^^R. Theicfoie m any such ^<is K lt ]lf and y ', I 10 
This is found to agree "\\cll with the valnesol ygol hydneel measnie- 
ment in an, oxygen, nitiogcn and olhei diatomic pei nuiin nl njises 
It follows also that the values of AT and A" (l deduced I nun I he I hi 01 \ , 
when expiest.cd m calones pei mol, aie foi all such ,ases 

, r 7x1 !)S5 



, T , 5 x ] 085 

and 7v = - - = I, <)(i;j 

From these figuies the following values of the specilie In-als aie 

* Ideal in tlie sense that the gas satisfies tlio oij[uatiou PV ItT nn.l ulxo |,1 1 at 
its molecules have no sensible onoigy of vilnation (Ait 17)) 



vi J INTERNAL-COMBUSTION ENGINES 211 

deduced for vanous diatomic gases, by dividing by the value of m 
nppiopiialc to each. 

(J lie. ul ik-d ipO( ilic licats m 



oi lb ' iloncs pei lb 
A',, K,, 

Oxygen 02171 IB/51 

NiUogi-n 02481 01772 

An 02408 01720 

,H7l 2481 

c oxido 02181 01772 



The obseived specific heals dil'lei a hi lie Irom Ihese, because 
he leases aic not ideal 

162 Summary of Methods of expressing the Specific 
ieats A shot t summai v of ineLhods of slal my K Jt and AT, in yases 
miy help lo a\oid contusion Kilhei of Hiese quantities mav be 
taLed as lollo\\s 

(a) In oi.unnu-Ctiloiics pei i^iamme-moh enle 01 mol, I lie 
.itiinme-moleenle 01 mol beiny <i mass equal lo in itammes, uh< ic 
'/ is the niimbei which e\pi esses the- moleculai \vei<>hl 

(b) In oiammc-caloiics pci- ntannne 

(c) In lb -caloi ics pci lb 

(d) In root-pounds pei cubic loot 
(r) In lool-pounds pei lb 

To com ci I 1 1 oni (ti) lo (h) 01 lo (< ) di\ idc b\ /// The mini be is in 
'>) and in (0 .ue the s.iim To con\eii horn (ft) lo (d) nuilliplv 
\ r .'5 ')() To con\( il I'lom (< } Lo (c) imilhpK by I 100 

The dilhu ncc behvceii A ; , and A',,, \\lucli is neaily constant in 
11 ^ascs, h.is llu lollou'inii \alncs 

fu (<i], 1 ( )S."i caloiies 

I !).S5 

In (b) and (< ), caloius 
in 

In (f/), T 71 l(K)l-})ounds 

'277!) 

In (c), " iool-|)ouncls 
m 

163 Measured Values of Specific Heats 1 1 is lo be expected 
r ial the actual specific heats of gases should shghllv exceed tlic 
alues calculated fiom the kinetic theory,, owing to the depart me 
i ical gases Irom the ideal conditions assumed m the theory 

Measurements of K } , by Rcgnault foi a number of gases gave 
nines which are somewhat less than the theoietical values, but 



213 THERMODYNAMICS |c-u. 

the method used by him is now believed U> have been al'lielcd l>\ 
a systematic error, the effect of winch WHS [<> m.'ike Ihi ineasined 
values too small, apparently by aboul "2 per mil * 

A modem measincmcnt of K for an by Swami |, by means <>[' 
electnc heating undei conslanLpussine, gives (when i educed lo I hi 
mean calory used in tins book) A',, -= 02M8 ealones pei uiul of 
mass, at C and OIK atmosphere, which is, us \ve should e\peel, 
slightly gieatei than Lhc theoretical nnmbei 

The conesponding value of K Avould be 17U5, hiking H lo 
be O 0688 as m Ait. 160 

f v has been dnectly measured by Joly for seveinl gases, by 
the device ot applying steam externally lo hi a I a coppci globe 
containing the gas, and compaiuig Lhc amoiml of sleam Iheirbv 
condensed on the suifacc with I he amounl oondcnsid on aiiolhei 
exactly similai but empty globe). His observed \nlne ol A",, loi 
air, under stanclaid conditions, I hat is lo say al 0" (' and one .1 lino- 
sphere when coriected for Ihe icviscd \alue ol Hie l.ilenl beal ol 
steam and foi the mean caloiy is 17'J!) incalones JK i uiul of mass 
This is in good agicement wilh Ihe value ol A ,, inlciied hoin 
Swaim's meabinemcnt of K v 'J'he mean ol Ihe two is \7'27 

Taking Swann't, and Joly's icsiills lonclhu, il niav 1 eoneludi d 
that the mcabiiicd ^aluc ol K loi an is .iboiil 17'J7 ealoiv pel 
unit of mass, 01 i 98 ealoi ics pei giamme-molc-euk , al <)' (' and one 
atmosphcic The same ligme may be laken ,is ncail\ I UK ol ollui 
diatomic pei manent gases (oxygen, inUogcn, e.nbome o\nh) 

Theie is conclusu ee\idcnce lhal Ihe speeilie lu-nl ol ll sc n.| S( s 
rises with the Icmpeiatine 'J'his poml, which is impoilaiil in 
relation to gas-engines, will be eonsuleied in Ihe in \| aihele 

S%vann also applied the melhod of eh clue healing lo (Uleuninc' 
K Jt foi caiboinc acid, and lonnd it lo be 02017 CM!OI\ pei <. Jm ume 
at 20 C and 02211 at 100 C.i} Jl we assume UK uleol <-haiin ( - 
with tcmpeiatme to be timfoim liom 0" lo 100", llu liguie lot A 
at would beO ]9C8, equn nlcnl lo 8 ()(i calories pei H ianiiii( -nidh - 
cule, in being <U The eoiiespondmg viilnc ol A' ; , is (, (>.S, mid y is 
barely 1-8 These results aie m I'air agieem. nl willi Ihose oblaiiu-d 
by Joly in direct measuiemenls of UK spec-die lual ol cnibtniic 
acid at constant volume. 

* SAwann,P/^ Tiam A, vol 210, p 231 Aim, U HJL ,,,, L of llm |{, ,i lrt |i AHHI 
Oommittce on Gaseous Explosions B A Hop lj>i)8. 
t loc.cil 

t Joly, Phd Tians A, vol 18J, 1801, p. 73. 
Reduced to mean calonoa 



<rc] INTERNAL-COMBUSTION ENGINES 243 

164 Variation of Specific Heat with Temperature It 
was pointed out in Ait. 21 Lhab a, gas might he perfect m bhe sense 
>f confoinung Lo Boyle's Law and to Joule's Law, so thai Lhe eqiui- 
.ion PV = IfT is stnclly applicable, and sLill have its specific heat 
/ary with Ihc tcmpeiatuvc, though Ihcie would be no vanaUon 
,vith the piessurc 

Aii3 r \ analion of specific heat wilh pressmc is due Lo inipcii'ecLion 
)f the gas In the permanent gases, Lheie is but little departuie 
Voni the equation PV= JIT except at picssincs niueh higher than 
hose thai me found in gas-engines Hence (hen specific heat is 
icaily independent of the piessiuc Even Lhc mixture piodueed by 
i gas-engine explosion, compusing some water-\apom and cai- 
)omc acid along with niueh mliogen, conforms to the equation 
y V RT neaily enough to allow Lhat equation to be applied m 
alculatmg the tempciatiuc Iioni Hie absence! piessme VlLhough 
he specific heat ol'such a nuvluic is undoubtedly somewhat gicatei 
I high piessiucs than at lo\\ pussines, the dilieience is not so 
onsideiable as to be taken mlo account in gas-uigme calculations 
On the othci hand, the specific heal of such a mivhiu, and ol 
lost gases, \aiicslaigcly \\iLh I lie tempeialuie. becoming gicalei 
s I he tempeiatuie uses, and the citect of llus on the \\oiking of 
as-engmes is fundamentally nnpoitant 

In nionalonuc t>ascs, such as aigon or iRlnim, iheie is hllle, if 
iiy, meicase of specific lical with use oi tempeiatuie, m diatomic 
ases such as o\\gen 01 mtiogen the mcieasc is consideiable, in 
asis of moie compk\ conslitiilion, sueh as the Uiatonne g.iscs 
[_,() and CO,, il is laigei still The pie'bcncc e>l these constituents 
i a gas-wig UK m:\tuie makes its laic of change of specific heat 
ith Umpei'ihiK gicatci than that of an The specific heal ol a 
ris-engme mivtmc at 20(H)C is about 1 S limes what it is at ()C 
An olnious result of the mcieasc of specific heal with tcmpeia- 
ue is that A\hen a elelimle quantity ol heat is given to a gas 01 a 
nxlme of gases as ioi instance by the explosion at constant 
L)lume in a gas-cngjmc, the use of lempcratme is less Ihan it would 
i: weie the specific heat lo keep constant, for as the gas geU hotter 
ich degree of use absorbs more and inoic of the available heat. 
r r hen the cxpeiimcnt is made of exploding a chaigc in the cyhnelci 
'an engine or in any closed vessel, it is found that the tempcratuic 
itually i cached is fai short of that calculated on the basis of con- 
ant specific heat, af tci making full allowance for loss of heat to the 
alls of the vessel. When this fact was first observed it was put 



244 THERMODYNAMICS [en 

down to impeifect 01 lathci delayed combusbioii of the chaise, 
the suggestion was that a laige part of the heat of combustion was 
developed gradually, in a comparatively slow process called " after - 
burning," \\hic\\ was supposed to confciniic aflcr Lhe explosion had 
spicad through the whole vessel and aflci the tcmperatuic and 
piessure had iisen suddenly to llie obsci vcd maximum The notion 
that there is any consideiable effect due to " after-burning" is now 
abandoned, and it is ricogm/cd that I he facts aic sufficiently ex- 
plained by icference to the incicasc of specific heat with use in 
tempeiatuie 

Measuiemcnts of specific heat, showing this mcicasc, have been 
made in vanous ways ' by dueet heating, up to high tempciatmes, 
under constant pressure, b> observing the use of tcmpciatmc in 
explosions, and also by a method due lo Clerk), in which the gas 
in an engine cylinder is successively expanded and compicssed 
several times while the valves aie kept closed In that piocess, the 
woik done by 01 upon the gas between any two points of the stroke 
is determined by measuring the aiea under I he indicatoi cuive, 
and is used as a basis loi reckoning the change of mtcinal energy, 
\\hile the change of tempeiatine is infciied fiom the cliange in Ihc 
pioduct of piessuie and \olunie The mUliod can be applied eithei 
to impiisoned an 01 to an exploded change' It is subject lo some 
uncei taint} in the estimate that has to be made ol the heat which 
is given to, 01 taken fiom, the gas by Lhe cylinder walls 

The results of these \aiious methods ol experiment aie not veiy 
accoidant Jn geneial the figme got lor the specific heat of a hot 
gas, when the heating is done by internal combustion 01 by com- 
piession, is gieatei than when the gas is heated unclei constant 
piessmc. All the methods arc liable to ciiois which arc difficult 
to allow foi They agicc m showing that Lheie is an impoitant rise 
in specific heat with tempeialmc, gieater m Uialomic gases such 
as watei-vapour or caibome acid llian m mtiogcn 01 an The late 
of mcicaseis probably not uniform in dialomie gases, it is certainly 
not umioini in tiiatonue 01 moic complex gases 

The results aie olten expressed by means of a formula implying 
a uniform late of increase, 

K v = (K V ) Q + at, 

* Particulars ol these, and a vulimblo diHCuasion of Lhe toaultg, will bo found in 
the Eepoi ts of the Butish Association CoiumiLLoo 011 CJasoous Explosions, fiom 19US 
See also Sir D Cleik's book on Tltc Gas, Paid and Oil Emjutc, vol i 

\ D CIcikjPjoc Hoy Soc A, vol 77 



vi] INTERNAL-COMBUSTION ENGINES 245 

Avhcie (A r ) is the specific heal at C , K v is the specific heat 
at any tempciatuie /, and is a coefiicicnb that is constant for the 
paiticular gas 

Similaily, K 1} = (K P ) Q + at. 

The coefficient a is I he same foi any one gas as in the formula for 
K v < since (assuming the law PV= RT to hold) the difference be- 
tween 7v ; , and K^ is a constant, independent of the tempeiatme 

Moic piobably, howcvci, I he uitc of mcicasc of specific heat 
with tempciatuie is not constant, and foimulas of the t} r pc 

K v = (AT,,),, + at + pl\ 
K 3> = (A%) () + at + Pi 2 

aie leqmied to cxpiess the i elation, especially in gases where there 
is much change 

In the absence of nioic exact data a hncai fonnula may scive 
Loughly, Ihiough a modeialr langc, foi an, nitiogcn, oxygen, and 
caibonic o\ide, namely 

A' t = I OS | 001 /, 

111 caloncs pci giainmc-molceule 

.For caibouic acid Langcn ' gnes, as applying fiom 1100 lo 
1700" C , a foiniulci (based on explosion cxpciimcnLs) |iu\ aleul lo 

K = (j 7 H- OO.V2J, 
ind foi waLei-vapom he gn cs 

A',, = 1 <) 1 -1- OOl'V, 

both m c.iloiics jici giainme-molec iilc 

Tlicic c'.in be lilllc doiibl, ho\\(\i'i, Uial a hncai I'oimula is not 
i e<i II y applicable lo I lu M' g.iscs I Inonnli ,in\ r \\ idc- i .ingt A leim in 
- is ictpiiied <is \\ell .is a Imn in /, cspeoulh al Icinpci.ihucs such 
is aic icaclied in y.is-engiiu's 

Results colic cltd b\ r I he Hnlish Associahon Commiltcc, for a 
Lypic.il gas-enoine nuvluic, \\ill be discussed in Ail 108, and li will 
jc shown thai I hey unolvc a loimula of Ihe Ij-pc 
K = (A'J,, + / -I- fit- 

165 Internal Energy of a Gas What we aic piachcally con- 
icincd AVI Hi in the gas-engine is not so much the specific heat as a 

* Langim Xcilt dca Vex nics d&ntacJiei Inyancine, vol 17, 1003, p d22 
| Piobalily lias IH too low In an ideal tuatomic gas A',, would ho 3K or 5 905 
Ai I 17 !) in water- vapoui it, should bo higher Values of Uio spcoifio heal ol watoi- 
found by Pioi in oxploRion ovpornnonLs (quoted m Raokiu's "Thonno- 
and Thermodynamics," Tin Gibson, p 72) irmko A',, noaily G 1 for t = Q 



246 THERMODYNAMICS [or. 

quantity closely iclatccl to it, namely the internal energy W When 
the charge is exploded at constant volume ils internal encr^v 
mcieases by the amount ofhcal developed, less what is lost In the 
cylinder walls. 

In adiabatic compression the gas gams internal ewii>v t <|m\ alent 
to the woik spent in compressing it, in adinbalie expansion i! loses 
internal energ} r equivalent to the work done Hehveen ,mv h\o 
points a and b on the cm ve of expansion 01 eomj)iessioM in .in .ielii.il 
engine, if theie is no combustion between ft and b, the equation 
holds 7T _ TT _i TI/ i n 

&a Jj J> + '"nb ~l <*:&' 

wheie E a is the internal energy at a, E b is tin 1 mlein.il c nei<>v -il />, 
W ab is the work done bv the gas in chan<>m<> fioin slnle a lo sl;de /;, 
and Q ab is the heat lost to the w.ills dninii> (hat ch.ingc of sl.ih 
If theie is any inteinal combustion between a and /;, LJCIK i.il ni'; 
heat repiesentecl by Q' ol , the equation takes ihe nion ^ciu-i.il 

f01 ' m * 



Hence if we know the values ot E loi all stales v\e m,i\, l>\ 
ing the indicatoi diagiam and so nic.isin nn }}',, I'oi .m\ slip, 
deteimme completely the liansl'ei of lie.il belween o ils ,iiid UK l.d 
A knowledge of the values of E for the woilunn <;,is I |IM>IIM|I<MI| 
a lange of temperatuie fiom sav 100 C 1 lo iiooo" (' | S MK M (,, 
of great piactical impoitance What is w.mled is ieiii\( S|I.)\M M ,, 
the relation of E to the tempciatuie f 

The relation between the mlcinal eneig\ K .md Ihe S | t die h, .,1 

is that ,,-, r , ,, 

(IE = A,,r/7 

Hence, at any tempcicitme, the slope of (he em\e ,.| l<] .,,,<! y 

, dE 
nam ely ^ measures K v , and E - JK v tlT 

If K v weie constant the enr\e of mteinal <nei..y would !., .1 
stiaight line and we should have 



,, 

Here t is the temperatmc on Ihe eenhgmde se.de, ,.,! || u - ,.,. 
stant of mtegiatmn is /cio if the us.ul eonvenhon |, ( . m | n|ll( ,[ , 



vi] INTERNAL-COMBUSTION ENGINES 217 

icckoning the cneigy of the gas from an arbitraiy staiting-pomt ab 
C. This of comsc docs not mean that a gas at C. has no in- 
ternal energy, but only that the staled value at any tcmpeiature 
is the excess above the value at (compaic Art. 60). 

If K v = (K V ) Q + at, 



Oi, if K v = (K ) -I- at + pi*, 

n lr , . at" # 3 
E = (K v ) t+ 2 +'- 3 

We may accoidmglv construct a cuive of E and t when nn cx- 
picssion foi K v is given, or conveiscly find an cxpie^sion for K v 
fiom a >n en cui vc of K tnul / 

Fuilhci, when the cinvc of E and / foi a gas 01 mixtmc of oases 
is diawn, the value of K v at any icmpcratnic is icadily found bv 
mcasuimg tlie slope of Ihc cin\e Iheic. Fioni llial K niav be 
deduced by adding Hie g.is-eonslanl R to A',,, namely 1 OS 5 if A',, 
is cxpicsscd in giamme-caloiies pei giammc-mokcult. (\it 1(>I) 
In llus wav Ihe latio y of K n lo K v m,iv be dcU'iinuud loi any 
lempciatuie 

1 66 Adiabatic Expansion of a Gas with Vai table Specific 
Heat In any o.i^ whose specific heat mci rases AMlh llu 1 ttnipcia- 
tuie, y is nol constanl but bcconus Ii ss ;is Ihe lc inpe i.iline us' s 
This is t in ob\ ions consc(inence of Ihe f.icl lh.il \\lnlc Ihe specific 
lic<ils mcicase \\ilh using Icmpeiahne llu diffeituee bt I \\een Ibeiu 
keeps eoiislanl, since Ihe nas shll conloiins lo Ihe equation 
l*V ~ RT Ilcnee in alleinpluin lodiawnn adiabalie eui\c lot Ihe 
expansion (01 conipiession) of such d gas, lluough any gn en laho 
of volinui's, by means of Ihe ee[iiahon 1*V J consl.ml, \\c have 
lo do wilh a eonlinuous vanalion in Ihe index; y. 

When we know the icla1ionof7 lothc lenipeialine we may obtain 
an appioxunnlion to Ihe euive bv subsliLuling foi the vanable 
index y a conslanl index >i, chosen so lhal the gas does nol on the 
whole gain 01 lose ho.iL in Ihe process Foi llus purpose an aACiage 
value ol y is guessed al and piovisionally used as Ihe value of n. 
The J'oimula PI'" = conslanL is llien applied lo com pule Ihe work 
done, and also Ihe iiiuil Lcmpeialure The nulial lemperaLuie is 
assumed lo be known, as well as Ihe initial pic^suie and volume. 
It is then seen whether the amount of work so computed agiecs 



248 THERMODYNAMICS [cir 

with the difference between the values of E for the initial and final 
temperatures If they agree, the gas has on the whole neithei gained 
1101 lost heat If there is any discrepancy, ib is to he collected by 
using a somewhat diffeient value ol n This process of tnal and 
erroi gives a PV curve winch does not exactly coincide with the 
true adiabatic cuive but icpiesenls it fairlv well It falls lathei 
too fast at fast, foi the value of y is less than n in the eaily stages, 
later it crosses the true cuive and finally lies a httlc above it At 
the beginning there is some slight loss of heat in the assumed expan- 
sion with constant n, towaids the end theie is a gain of heat which 
balances that loss The final tempciatuie in the assumed expansion 
is a little highei than in tiue achabatic expansion, and conse- 
quently the woik aiea is a little less 1 -. The process can be made lo 
gn e as close an appioximation to the tiue adiabatic cuive as may 
be desned, if we divide the \\holc expansion 01 compiession into 
seveial steps, and deal in this niannei "uith each step in succession, 
finding an appiopnate ^alue ol the index // foi each step 

When a foimula connecting the specific heat with the tempeia- 
tuie is established, a icLition between F and T dining adiabahe 
expansion can be obtained as (ollo\\s in leims ol the coefficients 
used in the foimula, and fiom that i elation the foi m of the adiabal ic 
cuive can be detei mined Let 

K L , = a -I- bT -|- cT~ 
and K ]t =-- a' + bT + cT-, 

T being as usual the absolute tempciatuie. 

Since in adiabatic expansion the woik done by the gas is equal 
to the loss of internal cneigy 

PdV =-dE=- K u dT 

. . _ (IT Pdl" 

Dividing by T, A, T + - ^-- = 0, 

T , dT RdV n 
or A T + -y- = 0, 

72/F being equal to PIT. 

But K v = a + IT + cT 2 and K = a' - a 

Hence a d ?- + bdT + cTrlT + (a 1 - a] ~ = 

* This piocess of appioxunating to the adiabatic cmvo is doscribor] by Ilopkin^on, 
Proc In-st Mcch Eng , Apnl 1908, p 443 



vi] INTERNAL-COMBUSTION ENGINES 249 

Integrating, 

rT z 

a loo t T + IT + 4- + (' - ) lop e F = const 
2i 

rr, - hT \ 

01 T"F"-e - -const, 

e being 2 71 S3, the base of the Napicnan logai ithms This equation 
connects T \vilh V in I lie acliabatic expansion 01 compression of Lhe 
gas The most convenient way Lo apply it is to \\oik out the value 
of Ihe cons I ant foi the initial volume and Icmpcialurc of Lhc gas 
then take another value of the tcmpeiatuic (]OAVCI foi expansion, 
highei for compicssion) and find V foi that, and so on. Bj diawmg 
a cuivc of V in iclalion to T it it. easy to find the tcmpcialmc foi 
any assigned ralio of expansion 01 compicssion, and then to mfei P 

PV 

by means of Ihe i elation -^-~ constant 

2' 

Since, by that iclalion 7' = PF a y constant, the above equation 
for adiabatic expansion 01 compiession may be put in this fonn, 

P"V" e' ~ ~= const 

")i, chminalmg V, P-"'T"e ' --const, 

which duecllv connects P with T But neithei of these altcinalive 

01 ins is so convenient as the fust foi di awing the adiabatic cui\c. 

167 Ideal Efficiency as affected by the Variation of the 
specific Heat with Temperature \Vc saw (Ail 1 r > r >) that 
he standaid of efficiency known as the "an standaid" assumed 
onslanl specific heal When Ihe woikmo substance is .1 t>as, 01 
Mixhne ol gasis, m \\hich Ihe specific heal me leases \\illi the 
em])eialme, I lie ideal enicicne> , \\hich the enomc would icach jf Ihe 
om])iession and expansion weic adiabalic and the explosion weic 
ompleled al Ihe dead-point without giving up any heat lo Hie 
ylmdc r walls, is neee'ssanly much less m consc()iience of the lowei 
Maximum e>l lempeiatuie The ideal efficiency with vanablc 
pecific heat ma}' be elclei mined Avhcn we can cxpicss /v,, as a 
nnclion oi Ihe Icmpciatmc, 01 when we know Ihe iclalion of the 
uLcrnal cncigy to Ihe lempcratmc, but it elocs ne>t admit of any 
implc cxjjrcssion It may be computed by woikmg out Ihe lelcal 
idicaloi ehagiam, with adiabalic curves, assuming a heat ofce)in- 
'iisliem appioj)iialc lo the given mixtiuc, and then comparing 
he woik which Ihe diagiam rcpicscnts with the assumed heat of 
ombusLion 



250 THERMODYNAMICS [en 

When we can expiess the specific heat as a function of the tem- 
perature we can calculate the ideal efficiency a^ follows Taking as 

befoie 



K p = a' + bT + cT\ 

assume an initial tempera tin c T at the beginning of compiession. 
This, which is called the suction temperatme, is gcneially taken as 
100 C. Then apply the adiabatic equation 



to find T l , the tempeiatme at the end of compiession Next calcu- 
late T 2 , the tempeiature aftei explosion, by equating the change 
of E to the heat given out by the binning of the chaige Then again 
apply the adiabatic equation to find T 3 , the tcmpeiatuic after 
expansion. When these four tempeiatuies aic known the coric- 
spondmg values of E aie determinate. The woik spent m compies- 
sion is E-L E Q , the work done in expansion is E^ E^, the heat 
supplied is E 2 E! Hence the ideal efficiency is 

E- E, - (E, - E Q ) 
E 2 - E, 

When this calculation ib made foi a typical gas-engine mixture 
the lesult is to give an ideal efficiency about 20 pei cent, less lhan 
the air standard, foi such latios of compitssion as arc in common 
use ' . We saw m Ait. 155 that with constant specific heat and thcie- 
foie constant y the an-standaid efficiency was 



A good appioximation to the efficiency of the ideal cycle with 
vanable specific heat is given by the cmpincal foi mi i Id 



Heie the index 3 ma}' be said to icpiesent a gencial average of 
Lhc values of y 1 throughout the action (assumed to be adia- 
batic) of a typical gas-engine mixtuic. (Compaic with Art. 168 
\\heie values of y aie gncn for vanous tempeiatuies.) 

* See Mr H E Wimpens' tex:t-boolc ol Tha Inlet mil Compilation Engine, Chap iv, 
wheie results aie given in detail, for various values of ? The specific heats aie 
theie expressed as linear functions of the tempeiature (the term oT & being omitted) 
For reasons \\hirh \vill appeal in Ait 168 the term m T- should be retained 



vi] INTERNAL-COMBUSTION ENGINES 25] 

It should be noted however that the ideal efficiency, for a mixture 
in which the specific hcaL vanes, depends not only on the latio of 
compression but also on the strength of the chaigc. A stiong 
charge gives on combustion a highei temperature than docs a weak 
charge, because of the greater development of heat, and it also 
pioduccs a mixture which contains a laigei proportion of waler- 
vapoui and caibomc aeid Foi both of these icasons Ihe influence 
of va nation in specific heat, in i educing the ideal ciricicncv below 
that of the air standard, is gi cater foi a sliong charge than for a 
weak one In an example given bv Ilopkinson *, where the ratio of 
compression was 6 37 and the an -standard efficiency was Lhcielbie 
0522, the ideal effieicncv (allowing for Aanation of spceilic heat) 
was computed to be 121 foi a mixture containing S 8 pci ccnl. 
of coal-gas before combustion, and 0301 foi a stiongcr mixture 
containing 11 '! pci cent. Fiom these figuies it would appeal IhaL 
a mixtme eontaining about 9', pci cent of coal-gas would ha\c 
an ideal efficiency 20 |)ci cenl less than the an sl.mdaid 

To dctcinnne the ideal c(hcicnc\ is a maltci of impoitancc 
because, by compaimg it \\ilh I IK clheicncv aclualh ie,ih/ctl, we 
aic able lo sa\ what is the niaigm of impio\emcnl foi leducmg 
the Iheimod) iianue losses that ocein in the aclion ol a icaJ engine 

168 Curve of Inteinal Energy for a Typical Gas-Engme 
Mixture The Untish Yssocialion Committee on Gascons Explo- 
sions QI\C in I hen fnsl Ucpoit (100S) ,i cui\ e of inleinal cncigvand 
Lcmpcialuic loi <i l\pical gai-cnginc mixlnie, namely the nu\tuic 
used b^ C'kikin Ihe cvpciimcnls to \\hich u I'eicnce has been made 
This mixluie was the pioducl of combustion of a cliaige ol one part 
by volume of coal-gas to about nine paits ol an, toclhci with Ihc 
bmnl gases in ihc cleaiance space it contained 5 pci ccnl by 
"volume of caibomc acid and 12 pci cent ol walei-\apom, Hie 
icmaming S3 pei cent being made up of mliogen and suiplus 
oxygen 

The cm vc, which is ic])iodueccl in fig 82, is based pailly on Clcik's 
expeiimcnls, m which the hoi mixtuic was expanded and compies- 
sed in an engine, paitly on earlv cxjilosion experiments by Mallard 
and Le Chatehei [ and Jatci ones by L'ingcn |, and paitly on duccL 
mcasuicmeuls of the specific heat of gases at constant pressme, 

* Proi Inil Mcih Kng , Apiil, 1008, p 425 
) Mallaid and Lo Cliatolioi, Ami dca Mines, Ib83, p 274 

f Langon, loc cil, , also Mittoil ilbcr Foisc/iiingsarbcitrii, font Vt> dcttlscfi Ing, 
Lleft 8, 1903 



252 



THERMODYNAMICS 



en. 



o 
o 
o 




9fnodioiu-9uiuivj$ jdd 



o 
o 
o 

CN 



O 

O 
O 

o 



vi] INTERNAL-COMBUSTION ENGINES 253 

by Holborn and ITenmno* The curve shows the value, in relation 
to the tcmpeiatuu, of E cxpiessecl in giamme-ciiloiies pei gramme- 
molecule that is to say in o rammc-ealories foi A volume of the mi\> 
tuie equal (o 22,100 cubic ccnLimctics at C and one atmosphere 
To reduce K to fooL-pounds per cubic foot multiply by 3 00 | 

The curve as on^inally given in Lhc Committee's Report stalls 
from 100 C , which is laken as the /eio in icckonmg E As here 
leproduced iL is extended down to C , and E is icckoncd from thai 
poml in accoidancc with Ihc convention alicady mentioned 

A careful examination of the cuive shows llut no foininla of the 
type K = (A",,),, -I- / will lit it K v must mciease slowly at low 
tcmpeiahues and fas lei at high tempeuiturcs to gi\c values of E 
thai will ai>iec with the cm\e It is, howevei, well rcpiesented by 
the loimula K = ,- 2/ + f) 00() j fji j + 000()0()2/ J , 

which couesponds (Art 105) lo 

K u = 5 2 + OOOSW + OOOOOOO/ 2 

Values of E measincd liom Ihe cuive and calculated horn the 
above loimula aie compaicd in the table below 



Intt'tnal Kn(i<iy <>J Gat*- Engine A 

J'J in caluncs poi litiiiniuo molcuilo 



Mioa^iiind hom 
IV nip ' tin ( iu\o 



200 1000 J0 r ><) 

K)0 21()0 2l()2 

1)00 ,{.{20 .J.JI8 

800 1, 1510 4/5 

1000 r >SIO f>8.JO 

1200 7200 7205 

I JOO 8(>,SO S(>72 

11)00 10,210 J 0,2 10 

ISOO l!,')20 ll,')20 

2000 13,720 13,720 

As was pointed oul m Ail K5, the slope of Ihe cmve mcasuies 
K v at any poml The initial slope, foi t = C , is 5 2 calones pei 
clcgicc The slope foi t = 2000 C is 9 32. Within I hat range the 
>peeilic heat has increased by nearly 80 per cent. At 2000 the gas 

* JLolborn and Honmng, Ann (hi Phi/s , vol 23, 1907, p 809 
| In the Report the volume of the gramme-molecule is given as 22,250 o cms 
ind this factor ns 3 96 Thoie appears to bo some enoi m both of those figures 



254 THERMODYNAMICS [en. 

contains 30 per cent, moic inteinal energy than it would contain 
if I he specific heat were constant at 5 2 

The specific heat at C agrees faiily well with the value which 
we might calculate from the known composition of the mixture, 
using the figures given m Ait 1 6 V namely C 7 foi CO_>, 5 9 for H,O, 
and -1 9S for the remainder of the mixture Since Ihe proportions 
by volume wcie 5 pei cent of CO,, 12 per cent of II 2 O and 83 per 
cent, of other gases, we should expect K u to be 

5x67 + 12x50 + 83x198 

_ foo = o 18 

If 6 1 weie taken foi II 2 instead of 5 9 the agreement would be 
exact (See footnote on p 2 l-l ) 

Values of K v , K }) (taken as equal to K h 1 985), and of y, lor 
this mixtuic at various temper atmes are given below 



Temp U C 


'' 


500 


100U 


15UU 


JOOU' 


K a 


5200 


5780 


OObO 


7840 


9 320 


K,, 


7 185 


7705 


S (5-15 


9825 


1 J 300 


y 


1 38 


135 


1 30 


1 25 


1 21 



The importance ol the t- tenn in Ihe foimula foi K is obvious, 
especially as alfcctmg the values at high lempci atmes 

The uses to which a cm\e connecting the inteinal cncigv wilh 
the temper aim e can be put in the analysis of an engine lnal aie 
illustrated m the next aiticlc 

169 Action in a Real Engine. Analysis of the Indicator 
Diagram The indicator diagram of an internal-combustion uigmi 
sei\es not only to mcasuic Ihe work clone, but to liace I he changes 
of temperature thioughout the cycle, piwided Ihe lempuatine 
at one point is independently ascci tamed, and piovidcd also Iheie 
is no leakage of the woiking substance Undei noimal conditions 
of good running., the leakage is negligible The gas itself serves as 
a theimometci, since T vanes as PV Thus if we compaic any IvVo 
points a and & between which there is no chemical change, 

/-> 77 p T r 

* a f <i _ *_i f _p 

T ~ T ' 
* a L l 

T P V 
from which 2' 6 =-"-. - 6 

* a' 

If however combustion has taken place between a and ft, we have 
to allow for the chemical contraction by introducing a factor a 



V,] 



INTERNAL-COMKUSTION ENGINES 



which cxpicsscs the ratio of the specific volume after combustion 
lo the specific volume befoie combuslinn This makes 

T p r 

7' = -' ?J _ '' 
aJt ,J\i 

Thus if we know Ihe "suction tempeiatme" T of the charge when 

compiession begins, it is easy to imd fiom I he indicator diagiam 

the lempciaturcs at any oilier 

points, such as the end of com- 
piession, the end oi' expansion, 

and the point of maximum 

piessure These Icmpciatuic's 

are of couise in all cases the 

mean temperature throughout 

the whole volume ol the gas 
Take foi example fig S3, an 

mdicatoi diagiam m one of 

Ilopkmsoifs tesls ' ol a gas- 

L'ngine \\ith a coni])iession latio 

of (j 37 In this instance the 

L>as-nuxtuic al Ihe beginning ol 

-ompiLssion had a lempeiatiue 
jf 100 C , and a piessiue ol 

L 1 7 pounels pel sq inch Itcon- 
amed 11 pci eenl ol coal-gas 
mel the chemical conliaclioii 
\as 3 pei cent , making cr -- ( )7 \\hcn Ihe ehaigc was igmleel 
he combuslioii \\as so lapid lhal Ihe piessiue lose, lo ils maxi- 
num beloie the |)iston had ino\ed pciccplil)l\ loiwaid L r siug Ihe 
ame sulhxcs as m lig 79, \\e see horn Ijic diagiam lhal P l al 
he end ol compiession was 105 pounds, /^ aflei explosion was 
S2 pounds, and J\ at the end of expansion was 10 > pounds, as 
iieasuicd by piolongmg Ihe expansion cuivc to tiie end of the 
tioke, beyonel Ihe point al \\hich the release-valve began to open 
lence (using absolute tempLralmes), 

117 x (> 37 



500- 



400- J 



300 1 



200- 



100- 





J'V r s J lndiuaLoi 




Du^um timiia(Jn> 
l')()s(ii l) ,,L ll ,,u,,) 



]> 1' 

1 ()' 



T 

rit * 2 

*- '1 ~r 



al\ 97 x 105 

'*- 1(J8 x '^ 5 x G J7 - 

_ ^_ _ 

* P/oo Jnst JUeUt Ena 1008 ID 420 



= 1080 C 



956 THERMODYNAMICS [en 

Or, calculating ducct horn the initial stale, 






T = 3 = 1,95 

J ff P 97 x 1 I 7 

Hence, if a cuive of 7? and T Ibi the mixtuie is available, it is 
easy to find the values of E at successive stages, and, by compamig 
them with the work done fioni stage to stage, to detci mine the heal 
oiven to 01 taken 1'iom the walls dining compiession, explosion, 

O 

and expansion 

In the example quoted, Hopkmson lound that the indicated woik 
lepiesented 33 pci cent of the heat ol combustion, and thai the 
gases at the tenipeiatmc oi icleasc (7' 3 ) cauicd oil ,'39 pci cent , 
leaving 28 pei cent as the net loss bv lachalion and conduction 
Most of this Jast item is taken up by the cnculaling walci of the 
watei -jacket which is used to keep the cyhndei cool enough loi 
lubucation. The 33 pei cent convcited into woik icpusenls an 
efficiency-ratio of S3, lor the ideal efficiency nndci tlu sc conditions 
ot compression and mixtuie stiength was 10 

With a weakei mixtmc, containing 8 5 pei cent of coal-gas, he 
found that a laigei fi del ion, namely 37 pci ecul , ol Lhe luat ol 
combustion Mas conveilcd mlo indic.itcd woik This concsponds 
to a InghcL clucicncy-ialio, namely 0<S7 As llopl mson obseixcs 
"the wcakci niixtines, in tidchlion to gi\mg a Inghci ideal c HICK no , 
come neaiei in pi ac lice io leahsing that ideal " This is bce.msc 
they lose iclalivcly less heat to Lhc walls dining t \plosion ,ind c \- 
pansion ''The diifcicncc is siifhcient to eounlcibalance <m mllu- 
ence lending the othci way, namely Lhc moic uipid combustion 
of the slionger mix lines " The less ia[)id combustion ol weak 
mixtures is appaient milieu mchcaloi chagiams it shows ilscll b\ 
the maximum piessuicoceuiung lalci and with amoie lonnded lop 
In extreme cases the combustion is piolonged Unoughoiil (he \\hole 
expansion stiokc, and I lie exhausl gases conlain unbmnl piodncls 

Incidentally these figures illuslialc the most obvious weakness of 
the oidinaiyinteinal-eombustion cycle that the latio of expansion 
is no gieatei than Lhc uilio oi compicssion Consccmcnlly Lhe pio- 
ducts of combustion aie chsehaigcd at a high Icmpeialme wilh much 
unutilized internal energy. Attempts have been made to avoid 
this loss, notably by Atkinson, who designed Joims of engine in 
which the expansion stioke was much longei than the compulsion 
stioke The advantage, m icspcct of thermal elficicncy, was con- 
siderable, but the complication of Atkinson's engines stood in the 



i] INTERNAL-COMBUSTION ENGINES 257 

'ay of their success, and in modern engines the theimodynamic 
d vantage of high expansion is sacrificed to mechanical simplicity. 

170. Measurement of Suction Temperature. In the above 

nalysis of the diagram it is obvious that the icsults depend on 

ic accuiacy with which the temperature is known after the admis- 

on valves die closed A icsistance theimomctci of ^ eiy fine wnc 

ill serve to measure it when furnished \\ith an electncal contacL 

i the shaft by means of which its indication is given only at the 

ghl moment in the stroke. But such a thermometer will not stand 

ic high temperature which is i cached in the explosion Messis 

illendar and Dalby 1 have devised an ingenious plan for ovci- 

>nung this difficulty In their device a fine wire of platinum foims 

LC theimomctci. It is fixed to a tube which slips llnongh a hole in 

ic spindle of the admission val\ e, and piojccts into the gas The 

be has a valve-shaped end which closes when it is drawn back, 

e wne is then scieened fiom tlic action of the gas, and no gas can 

cape. Bcfoic the end of the compulsion sliokc it is diawn back 

y means of a cam on tlic "salve-shall), and lemanis scieened 

nmg explosion and expansion. It is again piojcclcd into the gas 

foic the compicssion stroke begins The \al\c-slufl of lliccngine, 

nchiolales oncefoi each cycle of foui sliokcs, cauicsacam whicli 

mpleles the clectnc ciiciut of Ihc thcimomelei al the j)iopci 

ml, just when the admission ol the chaige is complete and com 

3ssion is about to begin 

171 The Process of Explosion Much light has been tlnown 
whal takes plate when a gas-engine nuxtnie explodes, by cxpen- 
i nls m which gas mixtmes have been exploded in closed \cssels 
constant volume, with devices lor icgislcimg the use of picssuie 
iclation lo (he time, anel also the piogiessive changes of tcmpcia- 
e al vanous points within the vessel The experiments of IIop- 
ison on explosions of coal-gas and an should be specially icfeiied 
in this connection |. 

_,ct an explosn r e mixture, homogeneous and at rest to begin with, 
ignited at any point A flame spreads in all directions from the 
nt of ignition, Uavellmg at a late which depends on the pres- 
e, so that each portion of the mixture ignites in turn, the most 

Proc Roy Soc A, vol. 80, 1907 See also the Seventh Report of the British 

jciation Committee on Gaseous Explosion (1914) 

B Hopkinson, Proc Boy. Soc A, vol 77, p 387, 1906, also vol 84. ra ICfi 



258 THERMODYNAMICS [cir. 

distant portions last When the initial piessmc is bhal of Ihe atmo- 
sphere, the flame may tiavel at the rate of only about five feet per 
second to beg in with, even in a nch inixtme such as one of gas to nine 
of air. The late depends on the richness of the imxtuic as well as on 
the piessuic in. a weak mixtme it takes much longci foi the ignition 
to spiead thiough the whole volume. When the initial picssure 
of the mixtme is high the ignition flame travels much fastci. 

The poition which is fiist ignited, close to the ignition plug, bums 
at neaily constant piessure, being sin rounded by a Luge clastic 
cushion of uniginted gas. Its combustion is practically completed 
befoie the piessme has nsen Then Ihe spicad of (he (Lime bungs 
more of the gas into action, the picssuie uses, and the portion 
which was hist burnt is compiesscd This compression is nearly 
adiabatic its effect is to laise Lhe temperatuie of that poi I ion much 
aboi* e the temperatuie to which it \\as brought bv combustion, and 
above the tempeia tine whichis reached in combustion b) the outlying 
paits of the gas, which aie compiesscd beloic thc\ become ignited. 

In Hopkuison's experiments a mixtuic. of nine pails ol gas lo one 
of air was, filed at atmosphenc picssuie in a c\ luuhical N essd with 
a capacity of about 6 cubic feel It was ignited b> an elecliic 
spaik at the centre, and dc\ eloped a maximum pussmc ol about 
SO pounds pei sq inch, uhich. was i cached a quailei ol a second al'lei 
filing The tcmpciatuic was obseived near the ccnlie- antl al olhei 
points On ignition Ihe tcmpciatuic at the ccnlu lose vei y lapidly 
to 1225 C , while the picssiuc icmamcd neaily constant In the 
latci stages of the explosion, \\hen the biunt gas al UK ernlie \\as 
being admbalically compiesscd, its tcmpeiatiue lose abo\c the 
melting point of platinum, piobably to 1000 This is a hi"he i tcm- 
pciatuie than was i cached in the outlying portions, \\luch xu-u- hist 
compiesscd adiabalically and then heated by combustion \\ hen the 
maximum picssuie uns i cached, the mean tempeiatuie infeiied 
from it was 1600. Hopkinson concluded that even m a vessel 
impei vious to heat, the poition of the mixhnc fhst hied would be 
hotter than the outlying portions by about 500, when the combus- 
tion of the whole was practically complete An interesting conse- 
quence, pointed out by him, follows from the fact that when the 
combustion is complete the gas is not in theirnal equilibrium 
Imagine the loss of heat to the Avails to be aucstcd while the burnt 
gas settles into equihbmun of tempeiatuie. If the specific heat were 
constant this settlement would make no difference to the mean 
tempeiatuie and therefore no diffeience to the picssuie But 



'ij INTERNAL-COMBUSTION ENGINES 259 

)ecause the specific heat of the hotter poitions is higher than that 
)f the cooler poi Lions, the tcmpeiatuie which the gas assumes when 
t is equally hot all through will be somewhat higher than the 
Dicvious mean, and theic will consequently be some use of pi ess ure 
is the gas settles into thermal equilibrium 

These cxpcnments go to show that Ihcie is no substantial 
imount of "aflcr-bmiung," and lhal the e fleets formerly ascii bed 
o altc'i-})iniiing are clue lo I he mcicasc of specific heat with tem- 
>eiatuie Piaclieally Hie full evolution of heal in eacli poition of 
he gas takes place at once when I he (lame i caches th.it poition, 
ml there is some dclav in completing the ignition of those poi lions 
/Inch aic in pioximily to the cold walls, especially when the 
nxtuie is weak 

In explosion experiments with weak mi\tmes the spread of the 
amc is much slower, so slow indeed lhal it is laigcly alfectcd by 
>m ection cuiients sc t up by I he ignihon of I lie gas ncaiest lo I he 
)<uk The gas in Hie up[)ci pail of I he vessel may bt completely 
mted \\hile the Io\\ei pail ol I h( \ ( sst I is still lull ol unbiimt 
>s By stilling I he conlenls ol Ihc \cssel so lhal the gases aie 
i motion when the spaik passes, a much moie i.ipid combuslion 
' I he whole can be seemed 

172 Effect of Turbulence This el ft c I of tmbiiltnce in 
omolmg i.ipid ignition ol live \\holt conlenls is fell, though lo 
less degiet, m shong mixlmts ,is \\tll as \\eak mixtmes II is, 
C'li i k has pointed out, an nn poi I an I f.ietoi in I lit 1 woikmg of an 
lei n.il-combiislion ( ngine \\'lu u a JK sh chaige is diaun in and 
mpicssed I he g<ists aie slill m moit 01 kss \iolenl nioliou <il 
c moment ol ignition Tins h<is I he yieal adxanlaj^e lhal coin- 
islion is lapitllv piop igalcd I hioughoul Hit chaige and (he ma\i- 
nm pi ess i ne oecius rally in I he expansion slioke. Ck i k obsei \ etl 
at when Hie e \plosi\t chaii>t in a gas-engine A\as nol died aftt'i 
e fust eompiession, bul \vas fired al((i llnee successive compics- 
'iis, so lhal (he liubulence set up on its cntiyhad lime m pail 

s\ibsidc, the process oJ combustion was generally prolonged, 
th the icsiill ot giving a lint diagiam and a wasteful action In a 
>h-specd engine the whole expansion stiokc may take only ouc- 
entielh of u .second, or less, and the explosion is over in a small 

ction of that time this would be impossible were it not foi the 
2ct of turbulence in causing the flame to spiead quickly through 

cylinder 



260 THERMODYNAMICS [en. 

173. Radiation in Explosions In closed-vessel experiments, 
the maximum of pressure is i cached a little before the combustion 
ib complete, for it occius when the late of loss of heal by ladiation 
and conduction to the walls just balances the late at Avhich heat 
is being geneiatcd in the gas Hopkinson has investigated the effect 
of radiation by comparing the late of cooling in an explosion vessel 
which was lined with highly-polished sihcr, with the Kite m the 
same vessel when its inner surface was blackened '. The late of 
cooling alter explosion was notably gi eater when the walls were 
blackened, and the maximum press me wo,s less foi chaises of the 
same composition The rate at which heat was lost Lo the polished 
walls was on the aveiage about two-thuds of the idle of loss to 
blackened walls It vaiied with the exact stale of the pohbhcel 
surface, which Avas nevei perfectly reflecting Ilopkmson conclu- 
ded that of the heat given by the gas to the walls of a blackened 
enclosuie dining the hibt quai ter-.seconel aftei maximum picssuie, 
at least 80 pei cent is radiant heat, and possibly a good deal more, 
foi the leflectmg quality of the polished walls may ha\e been mi- 
pan ed by the deposit of a film of moistmc al an eailv si age ol the- 
cooling Fiulhei experiments, in which Ihc \cssc'l was lillcd \\ith 
a fluonte wmelow to allo\v the laclialion to (all on an absoibiiig 
sci cen outside, confinned the view that lachafioii ace-omits foi moic 
than 30 pei cent of the \\holc heat loss. Its el fee Is \ve-ie slill per- 
ceptible aftei the tempciatine of the gas had Jallcn to 1200 

In latci mcasuieriients by \Y. T. l)a\'id I In 1 lotal loss b\ 
radiation attei the explosion of a mixtuic ol coal-gas and an in 
a closcel vessel, was lounel to be about 25 pei cm I oJ Ihc \\hole 
heat ol combustion The late at which ladianl eiiugv was 
emitted, thiough a fluonte wmelow, was giealesl a hi lie be-lou- 
the piessme i cached its maximum it Cell e>l'i lapielly as Ihc 
exploded gas cooled, bill ladiation could slill be delecle-d when 
the tempcialuic had fallen below 700 C., about a second allei 
the chaigc was fhcel | 

The eneigy of the laduition Jiom an exploded gas-engine 
mixtuic is due almost wholly to two 01 throe bands oi rays ol 
definite wave-length, coirespondmg to much slowei vibrations 
than those which pioducc the visible spec li um. The existence 
of these bands may be demonstrated by examining the heat 

* PWL Roy Soc A, vol 84, 1910, p 155 ISee also the Third Repoit of the Biitish 
Association Committee on Gaseous Explosions, 1010 

| W T David, P7iz7 Tiana A, vol 211, p 375 Phil May Fob I'UIiiuulJan 1920 



i] INTERNAL-COMBUSTION ENGINES 261 

r huh is radiated from a gas Maine when il is made non-himmous 
V using a 13unscn bmnci 

Experiments with such J lames show that when hydrogen is buinL 

j foim waLei-vapour niosl of the ladiant heat LhaL is given off is 

i a hand with a wave-length ol aboul 2 S/^ 1 , but some lias a longer 

'live-length also llial when caibonic oxide is burnt most of the 

uliant heat is in a band wilh a wuvc-laigth of about 1'i/j,, but 

>me is in two bauds whose wave-lengths ate about 2 7/j. and 

ehvcen 1 I- and Ifyi In one ease I he ladialion comes fioin vibrating 

lolcculos of II_,O, in Ihc olhei fiom vtbiahng molecules of (.'0, 

ll is also found lhal cold CO 2 absorbs sliongly Ihc radial ion tiom 

CO llamc, and \\alei-vnpour absoibs slionglv Ihc ladialion horn 

hydiogen (lame It may be concluded I hat the modes of fiec 

ibiahon of a molecule of cold CO, or wate-i-v apom ha\e penods 

Hicspondmg to the chic I \\a\e-Ienulhs which the gas gi\ cs out 

lien il is so violently agitated as lo become a sou ice of radiation. 

his happens \\hcn Ihc moleuilis aie foimed l>v Ihc coming 

)gclhei of then constituent aloms 

It is liulhei found | lhat a mixed 01 compound gas binning lo 

im CO_, and If/) gixts oul bolh wavt -lengths (I \JL and 2 S/t), 
id tluil Hie whole cncigN it ladialcs is Kju.il lo the sum of Hie 
K igies scpaialclv compiilcd lot I he molccuks of II 2 O and CO^ 
i.il aie foimid b\ ils combustion Foi e(|iicil volumes of !!_,() 
id C'O,, at the saint ll.imc Umpciahnc, the ladialion liom CO^ 
)|u.iis lo be .iboul "2\ liuus I h.il liom IF^O 

These icsiilts point let I he conclusion lhal \\heu a gas-engine 
ixluic is In eel, Ihe c'lu igv I h.il is ladiali d conn s almost cnliidv 

om molt cult s ftf ('()_, ,mel !!_,() in Ihe biiiul g.ises v ei\ hi tie ol it 

>mes liom Ihe mliogen 01 lhe i suiplus o\\gcn 

174 Molecular Energy ot a Gas Aceoiding lo Ihe kinetic 
icoiy of gases, the mlcin.il cne'i^v E ol a gas is made up of Ihe 
'inmumcable cncigies ol its mtilcenlcs, anel t.ich molecule m,i\ , 

ge'iiei.il, hav e communicable e'neig^y of Uiese- lluee kinds 

(1) Kneigy ol livmslalion of the molecule as a whole, 

(2) Eneigy of loUihon of Ihc molecule about an axis llnougli 
ils eentie ol mass, 

(,'J) Eneigy of vibialion 

* fj. staiKla for Jinlljonllis of a inolio TIio wave loiiglhs in tho vmiblo spoclium 
igo liom about 30 lo 11/j. 

\ R von lIclmJiolLx seo Hie Thud Koporl of Ihe BuLish Association ComnuUoo 



202 THERMODYNAMICS [en 

It is to energy of Hie first kind that the picssmc of I he gas is due 
The kinetic theory shov,s ' LhaL, in a gas foi winch PV = RT, the 
energy of translation is ]RT. The piessmc, in kinetic units, is 
mmieiicnlly equal to two-thirds of Lhe encigy of translation of 
the molecules in unit volume of the gas When Iho gas is healed, 
this encigy increases in dneei propoition le> T Hence \( nil Ihc 
mteinal eueigy of the gas wcie in this foim we should have 

E = \RT 

(leckoning E from the absolute zeio of lunperatnrc), and Ihc 
specific heat would be constant. K v would I hen be equal lo }]f, 
which would make K lt = r jR, and y = r \ oj 1 007. This is ncaily 
tine ol actual monafomic gases in such gases E consists cniirelv, 
01 almost entnely, ol cncigy of tinnslaiion of the molecules 

The second kind, energy of lotation, becomes an important pait 
of the whole when the molecule compnscs l\\o 01 moic aloms \Vc 
ma} conceive the molecule of a diatomic gas such as ()_, or N 2 to 
consist of paucd atoms held at a dcfunlc distance apail like the 
hea\y ends of a dumb-bell Such a stiucluie may, in the eouise ol 
its cncounleis, aequne cneigx e)l lotahon about any axis pei- 
penehculai to the line pining the I wo aloms, bill not about thai 
line In addition to its tlncc degiecs ol Cieedoni ol lianslalion it 
consequently has tttoty/fc/ji't 1 dcgices ol I'leeelom ol lolahon, hence 
fncmallaic effccli\e out of the six elcgiccs ol lieidom \\hich it 
possesses as a iigid body. 

According to the kinetic Lhcoiv the encounlcis bel \\een the mole- 
cules, when the gas is m a stead} state as lo picssiiu and lempeia- 
tuie, cause the cncigy of lianslalion anel lolalum, (]) and (2) 
togelhci, to become equally ehvielcel among as many ol lhese v six 
degiecs ol fieedom as aie elftcliye 

Hence in a ])eilect eliatonucgas, besides Iheeneigy of Iranslalion, 
vhich is \R r l\ llieic is .in amount of cncigy ol lolalion equal lo 
RT due to the two heedoms ol lotation, making ]RT in all foi the 
fi\e effective dcyices of I'le'edom. Consequenlly m such a gas, il 
theic wcic no energy except what is compiiscel m (1) and (2), we 
should find E=]RT, the specific hevt would be e-onslanl, K v 
would be 47?, A' 3) would be ]R, and y v\oulel be I 01 1 I 

These \ allies agicc \\ell with those Jounel in actual diatomic 
gases biich as nitiogen en an, so long as the gases aie ce>ld Uul, as 

* See Appendix. I f. 



vi] INTERNAL-COMBUSTION ENGINES 263 

we have already scon, Ihc specific heats become distinctly higher 
at high temperatures and y becomes less This means that, in 
addition to items (]) and (2), Ihcie is in these oases some energy 
of vibration (8), the amount of which is insignificant at low Icm- 
pciatnics, but becomes compaiaiivcly imporlanl when the gas is 
highly heated. It does not increase proportionally to T but in a 
moic rapid latio 

In Inatomie gases such as II 2 or C0 2 , and in gases of a more 
complex constitution, thcie arc Ihiee effective freedoms of rotation 
as well as llnec (Vcedoms of translation, making SIK in all, between 
which the cncigy compiiscd under items (1) and (2) is equally 
shaied Thus items (1) and (2) account foi an amount of cncigy 
equal Lo ( \RT If theic wcic no more, name ly no encigy of \ ibia- 
tion, (he specihc lu il would be constant, K H would be 3A\ K fl 
would be I A', and y would be ', 01 1 ,'333 

In walei -vapour and caibonie acid the \aliu of y c\ en al low 
Icmpcialmcs is less than 1 333 in walci-\apom il is about 13 
and in caiboiik acid il is a little lo\\ei (\it 103) Fiom this, and 
also fiom the lad thai modcialc healing c'onsidi i.iblx laises the 
spec'ilic heat, it m<i\ r be m lined that even at low lcni|)i tal mcs I hi 
moliciilis of these gases h<i\ r c some eneig\ ol \ibiahon [Is pio- 
poi lion lo I he \\ hole em igy is me i east d by hi a I ing the i>as 

The amount b\ \\huli the cnugv ol \ibialion augmuils Ihc 
specific heat in an\ g is ma\ be mleiied hoiu I he \ alue ol y il \\e 
assume the gas-law P\' = RT Lo appl\ Take loi mslance a tn- 
alomic gas A,,, il then \\eie no \ibialion, would bi {/', li I nR 
IK (hi amoiinl bv \\liuh \ibi.ihon.il cn< m\ mcieases il Thin 

K r - (.3 |- n) If, A,, - (I -I n) It, 
fiom which y -= ( I | )/(3 -|- //) 

Suppose that y has I he \alue 1 .30 instead of 1 ,3.33 I Ins makes 
n J, and the spieiln heal A' 7 , is thc'icfoie 10 pei cent giealcr 
because 1 ol vibialion The value of n mcieases with the lempcia- 
luie At the lempeialuic i cached m a gas-engine explosion y for 
CO 2 is piobably nol much moie than ] 1 1., which would coiiespond 
lo a spce-itie 1 heal approaching (>A' (See Ail 22I-.) 

The phrase "c'licigy of vibration" is to be understood as including 
all the kinds of eneigv which the molecule may acquire, in the course 
ol its encounters with other molecules, except eneigy of rotation 
as. a whole and energy of translation as a whole All such forms 
of energy arc internal to the molecule rtsclf they may be due Lo 



264 THERMODYNAMICS [en 

relative motions of its paits 01 to electucal clistiubanccs within it, 
01 within its atoms. It is lo encigy of vibiation I ha L the radiation 

9t 

given out by a heated gas is altnbntcd. 

When a gas-engine mix tine is fiied the energy generated by the 
explosion is at first concentrated in the newty-formed molecules 
of C0 2 and H 2 and spreads to the other molecules as a lesult 
of enconnteis. We may conjectuic thai Jl is at fust mainly 
vibrational, and the enconnlcis transform part of it into encigy 
of translation. It is eleai that the newlv-foimed molecules possess 
much moie than then noimal piopoilion of energy of vibrdlion, 
much moie, that is to sav, than Lhev Avould possess if the buint 
mixtme weie kept wiihoul loss of heat long enough to let 
equilibrium be attained between the different kinds of encigy, 
or were le-heated to the same tcmpeiatuie altei being cooled 
Some time, peihaps only a veiy shoit time, must elapse bcioie a 
condition of equilibiium is icached. II the gas weic enclosed, aftei 
combustion, in a vessel impeivions to heat, \\hi\c this process is 
going on, the eneigy of tiansLilion uonlcl incicase at the expense 
of the eneigy oi Mbiation, and the tcnipciatme would Ihcicfoie 
use though the total eneigy undergoes no change So fai as il goes, 
tins piocess of attaining equihbiiuin lias an c flee I like conhniicd 
combustion or "aftei-binnmg." The time taken to reach cquili- 
bnum is not known. If the pioccss is not \eiy soon oonij)lcled jl 
may account foi the fact that mcasuicmcnls ol specific lie.it made 
by means of an explosion in a closed ACSSC! gi\c \alues SOUK \\hal 
gicatei than those that aie got when the gas is healed in oilier 
ways 

It has been suggested that the molecules of a hoi gas emit 
lachation mainly when they unckiyo stiuctiual change If llns view 
be eoncct vc should expect a gas mixlinc lo rachale moie eneigy 
immediately aftu explosion lhan when it is maintained at Ihesame 
tcmpcialine, 01 ic-healed to the same tcmpeiatuie aflei cooling 
Hopkinson's and David's experiments show lhal in an explosion 
I he gas continues to rachale for a second 01 so alter maximum 
piessuie This may only mean that the special vibialions (special 
m violence 01 in kind) that aic set up dining the a,ct of foimalion, 
to which ladiation is ascnbed, subside lalhei slowly Thcie is 
in any case an action going on m all hot gases, Ihat Lends to 
maintain such vibiations, namely the bicakmg up of some mole- 
cules by exceptionally violent encoimteis, which is called dissocia- 
tion, and their subsequent re-foimation. 



vi] INTERNAL-COMBUSTION ENGINES 265 

175 Dissociation In airy gas, however homogeneous, and at 
any lempeiatuic, the molecules at a given instanb have widely 
vaiious speeds Some ol' the encounters may be so violent as lo 
bieak up compound molecules, sepaialing them into paits which 
after a time meet ficsh pailncis and le-combme The probability 
of such chsmptive encounters is ob\ lously grcatci the hot lei 
Lhc gas is In a hot gas in equilibiium, a pioccss of dissociation 
and ie-combinalion goes on continually, to an cvlenl depending 
on the tcmpcialme, with the result lhat al in\\ instant a 
certain piopoilion of I he gas is in (he dissociated stale The 
piopoition dissociated depends also on Hie picssiue, at high prcs- 
suie it is less than al low picssure, foi the same Icmpciatmc 
Accoidmg to measmements by Ncrnst and othcis the amount 
of II 2 O dissociated, undei a piessure of one atmospheie, is baicl> 
2 pci cent at a tcmpciatmc ol 2000 C , baich 1 pei cent at 1800, 
and 002 per cent, at 1227 C Al a pitssinc ol ten almospheies 
these numbcis aie about halxcd In C'O 2 at one almospheie, the 
piopoition dissociated at ](>f>0 C isaboul 1 pei cent and at 1200 
about 03 pei ccnl Al such tcmpeialincs Ihcic is })iobablv no sen- 
sible dissociation in mliogcn These dailies au ope n lo some doubt ' , 
but it they can be accepted as applxmg to I IK conditions ol a gas- 
engine nu\line a fit i explosion (conehlions \\lueh <ue not those ol 
c quih bi mm) il appeals that dissociation pla\ s no conside lable pail 
in that action So lai as it has <my ellecl it letluecs xeix shghll}, 
I he chc mica 1 con 1 1 action, by si i bsl i lu ling some molt ciiles ol II, and 
0, loi molecules of ILO, and some molecules of CO and ()_, loi mole- 
cules ol ( O 2 , (01 Ihe same icason il i educes slightly I he immedialc 
r lcvclopnienl t)l Iheimal eneigv, leaving a sm.ill piopoition of the 
ivailable chtimc<il encigy of the g.iseous fuel lo be ele\ elopeel latei, 
is the piopoilion of dissociated molecules diminishes with hilling 
empcialine The elkcl is Iheiefoie cqunalenl lo a continued 
'ombuslion 01 '.il'lti -binning " Ol, if we icgaid I he whole thcimal 
MK'igy <is being devc-loped at once and then a small poilion of it 
is being absoibtd by the bicaUmg up of some of the molecules 
n consequence of (hen cneoimlcis, the effect of dissociation is 
ndistmgmshable from lhat of inci eased specific heat 

* Sou tho yocoud Koj_>oiL of tho Uutuli AflsociaUoii (Joniimltoo on Gaseous E\- 
l')0 ( ) 



CHAPTER VII 

GENERAL THEHMODYNAMIC RELATIONS 

176. Introduction In the eailici chaptcis but JiLtlo u<-c was 
made of foimal mathematics in introducing the vcadci to the 
fundamental ideas of thennodynamics To most sludcnls Iheic 
is an achantagc m having these ideas so piescnled then physical 
significance is moie likely to be appreciated Once lhal is giasped, 
the btudent may pioceed to a moie mathematical heal men I with 
less ribk that the real meaning of the symbols uill be obscmcd m 
the analysis. But ti niathcmalical ticatmcnl musL be icsoilcd to 
if we wish to cxpiess with anything like completeness (he lelahons 
that hold between the vaiious piopcilics of a fluid 

One ol the uses to whicli these illations can be pul is in darning 
tables 01 charts of the piopcities of the fluid Hy then aid such 
tables can be compiled fiom a small mimbei of expciiinental data, 
and the experimental data themsihes, as well as the numbeis com- 
puted fiom them, can be tested foi lliermod> namic consist ency 

The purpose oi this cha|)tci is to show ho\\ the methods of the 
difleiential calculus may be applied to obtain, by mliiencc 1'ioin 
the Fust and ^Second Laws of Thcimodynanues, ceitain geneial 
iclaiions between the pwpcilics ol any fluid With some of these 
results the icadci ol the eaihei chapters is alieady acquainted 

In the next chaptu some applications of Ihe-se gnu tal lelalions 
to paiticnlai substances will be considered, including impelled 
gases, 01 ical fluids in the stale of \apoui In pailiculai il will be 
explained how Callcndar has emplo\ed them in calculating his 
tables of the piopcities of steam. 

177 Functions of the State of a Fluid. Assume lhal we 
aic dealing with unit mass of a homogeneous (hud As was pointed 
out jn Ail. 75, the six quantities named theie, /*, F, T, E, f, and 
</>, aic all functions of the state of the fluid, that is to say their value 
depends only on the actual state. When the fluid passes in any 
manner fiom one state to anothei, each of these quantities changes 



en vn] GENERAL TIIERMODYNAMIC RELATIONS 267 

by a definite amonnL which docs not depend on the nature of the 
operation by winch the change is effected, but only on whal Ihe 
state was bcfoic and Mhal it is after the opeialion has taken place. 
Tins fact is expiessed in malheniatieal lau^uaye by saying lhal the 
differential of any of these quanlihes is a. u peifcct" diffeicnhal. 
Oilier qnanlilies nnohl be added lo the list, which aic also functions 
of Ihe slate of Ihe fluid, such as the quantities G (01 , which is -- O) 
and i/r mentioned in Ail 00. 

In whal follows il is to be nndeislood lhal 7' means (as usual) Ihe 
absolute tcmpcialuu on the the rmodvnannc scale (Ail l>2) 

(1Q 
We defined the entropy </> m A.il 1 1< by the equation cty in 

a icversible opeialion, and the fact that cf) is a function of the slate 
was pioved URIC- as a consequence ot Ihe result lhal I = loi a 

rcyeisible cycle, a icsiill which follows from the Second Law of 
Thermodynamics The Second La \\ is Ihcu loie imohed in dialing 
(f) as a function of UK stale Ih nee Ihe lad lhal <lr[) is a peifctl 
difl'eiential is sometimes spoken of .is a inalhemalic.il expicssion 

ol Ihe Second Law II isnnpoilaiil lo notice lh.il \\hile , which 

is d([), is a pei led difleienli.il, <IQ itself is nol a peiled dilleicnlial, 
loi Ihe tnnounl of heal m\ol\ed in a change is nol a fiindion of 
Ihe state alone When a substance changes lionione stale lo .111- 
otlui, 1he amount of heal I iken in depends not simplv on wh.il 
Ihe l\\o sUiles <ne, bill also on Ihe nal me ol I he opeialion I >\ which 
Ihe change eucms Foi^lhc same icason, il \V icpiesinl the \\oik 
done dm inn \\ change ol stale, till is nol a pi i IV c I dilleicnlial. 

Since R, 1\ and J r aie- all lunclions ol the stale, il follows that 
the lolal heal /, Mhich is equal lo 1C + L'}'\ is also a hind ion of the 
stale Anel since 1 T and </> aie i also hind ions e>l Ihe slate, it follows 
lhal llus is also line 1 of, which is / 7'c/>, and ofj/r, A\hie'li is 1C - 7Y/> 
Hence ill, dL, and f/(/f, as well as r/ f />, ilK, <II\ <\V and tIT, aie pcifed 
differentials. 



178 Relation of any one Function of the State to two 
others The stale of Ihe Hind (assumed to be homogeneous) is 
coni]iletcly sj)cci(icd when any two of the .functions of the state are 
known. Any third function is then determinate, lh,it is to 
say, it can have only one value in any paiLicular substance. 
Thus if any two functions (such for example as the pi assure and the 



208 THERMODYNAMICS [en 

volume) be selected as "independent vanablcs," by icievence lo 
which the state is to be specified, then any I hud fiincUon (such foi 
example as the tempciaturc, or the total heat) may be lepioscntcd 
in relation to them by the famihai device of di awing a figure in 
which the two functions selected as independent vai tables aie i epi e- 
sented by lectanguLu coordinates X and F, and the third function 
is rcpiesented by a thud cooidiuate Z, pcipenchculai to I he plane 
of X and Y. This gives a solid figure, the height ol which shows, 
for any given state of the substance, the value of I he function Z in 
iclation to the values of the functions X and 1" which set NO to 
specify that state The surface of such a figiue may be called a 
thcimodynamic surface. 

Suppose now that the substance undeigoes an mliiulcsiinal 
change of state, so that the independent vanablcs change bv dX 
and dY lespectnely That is to say, \ve suppose X lo change to 
X + dX and Y to change to Y -h dY. Then Ihe tlnul luuclion 
changes fiom Z to Z + f/Z, by an amount </Z which may be c\- 
piessedlhus dZ=MdX + NdY (1), 

wheie M and N aic quantities depending on the iclalious ol Ihe 
functions to one another, and are thctcfoie also lunclioiis ol I ho 
state. 

This expression applies whethei bolh luncLions X and V \<IIN F , 
01 only one of them If X vanes bnl nol F, I lion dY - and 
dZ = MdX sumlaily if I" N aucs bnl nol X, dX - and dZ - XtlV 
Hence 



In this notation, ( ^] means the nilc of vanalion ol Z wilh 
\dA. i j 

respect to A" when Y is constant In the language of UK calculus, 

( -- v ) is the pailia! diffcienhal cocfTicicnt of Z wilh rospool lo 
V aJL / Y 

X when F is conslant, and (,} is the pailial difloicnli.il co- 

\dYl x 

efficient of Z with icspect lo F when X is conslant 

We might legaid the change of Z as occulting in two slops In 
the liist step suppose X lo change and F to keep couslanl The 
coriespondmg part of the change of Z is MdX, and /I/ is Ihe slope of 
the thermodynamic snifacc m a seclion-plane ZX. In the second 
step X is constant and F changes The corresponding pail of the 
change of Z is NdY, and N is the slope of the thermodynamic 



vir] GENERAL TIIERMODYNAMIC RELATIONS 269 

suifacc in a section-plane ZY. The whole change of Z is the sum of 
thcbe I wo par Is, as expiessecl in equation (1). The slopes along the 
two section-planes aic expiessecl in equation (2) 
Combining these equations we have 

dZ = ( d ^} dX+(~Pl dY (3) 

\fl\Jjf \dYJx 

These equations apply when X, Y, and Z aie mtcipieted as any 
tin cc functions of Ihc stale of a fluid Thus, foi instance, if we think 
of a small change of state in which the tempeiatuic changes fiom 
T to T + dT, and the piessinc fiom P to P -|- dP, the consequent 
chance of volume \\ill be 



'/' 

Sinnlaily, il Ihc \oliimc and picssuie change, Ihe consequent 
change of Icinpcialuic is 

'<rr\ 



Oi again, the change ol enliopy conscc[iient on a change oi tcni- 
pcialme and picssuie i^ 



and so on ll \\ill be obvious lhal a. \ c i \ Lngc iiinnl>ci of sinnlai 
cfjiuilions inighl be \viilUn oul, cadi using one pan ol lunclions 
of Ihc sl,il( as indc pi ndi'nl \ <iiiablcs,jind c\|)icssing in I cm is of then 
\analion UK \aiialionol sonic Hind Innclionol the slalc These 
me nieic'l\ lomis ol I hi gcncial c<[iialion (JJ) 

Rehn mug now lo Ihc gcncial Conn in -Y, Y, and Z, suppose a 
small change ol slale lo occm ol such ii chaiaclei LhaL Ihc function 
Z undeigocs no change In thai special case dZ = 0, Ihe steps 
AldX and A^/rc-anccl one anolhci Coiiscqucnlly 



dx. 

when (IX and dY au so iclaled lhal theie is no va nation of Z 
Hence the gcncial conclusion follows that 

dZ\ dZ\ dY 



This relation between the three partial differential coefficients. 



THERMODYNAMICS [en. 

holds, in all circumstances, for any three functions of the state of 
any fluid. It may be expiessed in these alternative forms 



'dX\ /dY\ dZ\ 






=- 
(dYj z ,dY 



Returning now to equation (1), 

dZ = MdX + NdY, 

the pimciples of the calculus show that when dZ w a pajcct dijfci 
ential, but not otheiwise, 

(dM\ __ fdN\ 
~ 



In dealing with funclions which depend only on I lie actual state 
of the fluid the condition that dZ is a peilect diffcicnlial is satisfied 
and consequently equation (5) applies. We shall see umncdialelv 
.some of the lesults of its application 

179- Energy Equations and Relations deduced from them 
Considei now the heat taken in when a small change of stale 
occuis in any fluid Calling the heat dQ uc have, by thcVnsL Law, 

dQ = dE + clW ( (J ) ) 

wheie dE is the gam of internal eneigv and rf/f is Ihc woik which 
the fluid does tluough mciease of its volume Since dW = PdV the 
equation may be wnttcn 

dE = dQ - PdV ( 7 ) 

Heie and in what follows we shall assume (hat quantities of heal 
me expiessecl m work units This simplifies the equations by allow- 
ing the factor J 01 A to be omitted. 

We aie concerned for the piesent only with icvcisible opcialions 
In any such opeiation dQ = Tcfy, hence 

dE = Td^ - PdV . (8)> 

Again, I^E + PF, by definition of 2 

= dE + d (PV] 

- PdV + PdV + VdP 
+ VdP . 



vn] GENERAL TIIERMODYNAMIC RELATIONS 271 

Again, = / T<j>, by definition of *. 

Hence f/ = dl d(T<f>] 

r fif1fL \ VfJP (l^fJrh -t- t\f] f ] 1 \ 

= VdP - <j>dT . . (10). 

Again, i/j = E 7V/, by definition of iff. 

Hence fty - dE - d (2V/,) 

= Tdtf> - PdV - (Tdcf> + falT} 

But dE, dl, d'(,, and difj aie all perfect differentials Hence, 
applying Eq (5) in tuin to Eqs (8), (9), (10), and (11) we obtain 
at once the following four relations between paitial differential 
coefficients 

fdT\ 



(II) 



" 5) 

Those aic known as MaxuclTs ['out thcnnoch nanno itl.ilioiis 
Expiessod in woids, the lnsl one nic<ins lh<il when <lll^ ^ Hind cx:- 
pands ctdial)aliealh r (f/> eonsl ) I he i.itc <it which UK lempeiatme 
/r///v pci mill mcKcisi of \ohnnc is K[iial to the Kite at which the 
piessmc would i is( , pei mill inciease ol' cnliop\, if the fluid weic 
heated at constant \olunie The second niediis lhat \\hen a (hud 
is eoinpiessed aduibalicallv the Kite a I which its tempeiahne uses, 
pci unit inciease of piessmc, is equal to the iatc at which the vol- 
ume would mciCriso pci unit increase of cntiopy if the fluid were 
heated at constant piessuio The thud means that when a fluid 
is heated at constant pressure, the rate at which the volume m- 
ci eases with the temperature is equal to the rate at which the 
cntiopy would be reduced per unit increase of pressure if the fluid 
were compressed rso thermally The fourth means that when a fluid 
is heated at constant Volume the rate at which the pressure rises 
with the temperature is equal to the rate at winch the entropy 

* For the sake of symmetry (", which 18 - &, is used horo rather than G 



272 THERMODYNAMICS [en. 

would increase with inciease of volume if the fluid were expanded 
isothermally 

The following furthei iclations are immediately deducible from 
Eqs. (8) to (!]). Taking Eq. (8), imagine the fluid lo be heated 
at constant volume. Then dV = and dE = Td(f), hence 



_ T 

til * 

d<f>Ji 

Again, imagine the fluid to expand adiabalically. Then dfy = 
and dE = - PdV , hence 



JVJ* 
Similaily fiom Eq (9) we obtain 

(dl\ ,'dl\ 

in) = J-> anc l r.J 

\dtpj j> \drj 

from Eq (10) (7) = V> anc ^ ( /7 ,) = 

fiom E, (II, (^--P.-l^.),- 

Collecting these results, 

'ill \ . ((IK 



1 80. Expressions for the Specific Heats /v',, and A",, In 
general the specific heats of a fluid aie no I constant, I hoy are 
functions of the state of the fluid We shall piocccd lo find 
differential expressions connecting I hem with the tcmpeiatme, 
volume and piessure Such expressions enable other pioperlies 
to be calculated when the i elation between T, V, and P is known 

Considei, as befoie, a small change of state during which the fluid 
takes in an amount of heat dQ while it expands in a leversible 
manner. Its entiopy accordingly increases by an amount d(f> such 
that Tdcj> = dQ. Its temperature changes from T to T + dT and its 



[i] GENERAL THERMODYNAMIC RELATIONS 273 



-from Fto V + dV Take, in the fust place, tlie tempera tine 
id volume as the two independent vaiiablcs b> means of which 
ic state of the fluid is specified The change in any third quantity 
ay be fc> bated with reference to the changes in T and in V Thus 
ic hca t taken in may be written 

(IQ. = K v dT H- IdV . (20) 

-- fdQ\ 

ere A- V9 which is the specific heal al constant \olnme. is .... 1 
' \dTJr 

, , , , , fdQ\ 

id / is o symbol for ( 



'T 
Since rf(2 = Td 



it by ISq (15) , . 

^'" /a 1 



-rffi ()- 



<1Q = K,,(IT + T ., r// r 
V///( 

l^olh sides by 2 1 , we have 

f/r/ = ,-." dT + I } <IV 

iis is ix peifcet dilfeicnlial, and thcieloie, b\ r Ivj 

i A '.- = ( f/ 
' 7 . 7' V^/7 1 

'r/A r , 
nee 



is is !> ii impoilanl piopc.i I y of A.",, 

To ol>l *nn a eoiiespondino'piopcilv of A r y ,, lake [lie lenipcialme 
1 p res .-sine as I he Lwo independent \tuiabks and express Ihe heat 
;en m ^v^lh leferenee lo I hem The. heal taken in, ilQ_, is I he same 
before;, beino still equal to Tdc/j We may wnlc 

dQ = K P dT + l'dl> (21) 

le K 3> , which is Ihe specific heat at constant, picssmc, is 



., , ancl Z' is a symbol for ., ) 
f IP \(ii IT 



jincc 



274 THERMODYNAMICS 

fd<f>\ 
But byEq (]4) (dp) T = ' 

Hence /' = T ( -y, 

and dQ = K v dT T 

Dividing both sides bv T, we have 



21 *" 
\tlJL J 1> 

And by Eq (5), since this is a perfect diffemif ml, 
A\ ?J - ( d \ ( <l} '\ 

jTp / ~~ni I 7'/' / ( ./'/i / 

^IF) T \(ll //' \(ll J j> 



T \ dP ) T 
01 l-rl =-'!'[ 1 (J7) j 



thepiopeitj of /ir^coiiespoiicliny lo lli.U ol A', in Kt 
Fuithei, fiom Eqs. (20) and (21), 



01 (/v j( - K u )dT = 

By wntmg dP = it ibllows LhaL 



K K l dV \ 
*- A = f ' 



Or by writing dV = 0, 

/ ( 
3> **- V ' ' I /r/i J 

V/77 , 

By Eq (21) or (25), eiLhci of lliese gives I Ins im,><>,l..ul rvp,. s . 
sion for the diffeience between the I wo spt-oilie heals, 

l' V 'r( (U ^ 



And since by Eq ( X) 

( dV \ _ (M'\ f'l f> \ 
\dT) P (tll'Jr^ir), ' 
tins result may be writ leu 



q (M a ) rt wnbe.KW, thai AT,, um-r l,,l,ss H.an A',,. 
for Jj, ls ^senbally negative, ,, IC ,V,.SP ol |,K. sslm . 



vii] GENERAL THERMODYNAMIC RELATIONS 275 

decrease of volume in any fluid, and Iheiefoie the whole expiession 
on the light is positive. Accoidmgly K D is always greater than 
K v , except in the special case when one of the factois on the nght- 
haiid side 11 equal to zeio, in which case K p is equal to K v This 
is possible in a fluid which has a temperatme of maximum density 
(as waLci has at about 4 C.) At the tempeiature of maximum 

density r , J = 0, and consequently at that point K f -K v = 0. 

Return now Lo Eqs. (22) and (26) In heating at constant 
volume dV = 0, hence by Eq. (22) 



In beating at constant piessure dP = 0, hence by Eq (20) 



K '- T ^) P < 80 > 

In a.n achabatic opeiation fZ</> = 0, hence by Eq (22) 
K t , ,dl\ _ _ tdP 

undbvEq (20) T j) dp),rC!r)p (32) 

Fin I her, by Eq (I h) 



,* K,fdV\ 
K,,fdT\ K\<1P)* 



T 

A, fdl r \ ,dl 



This is Lhc lalio usually called y. 

Thus in I he idiabatie expansion of any fluid the slope of the 
IT IIIK is y limes its slope in isotheimal expansion, 

(IP dP 



181 Further deductions from the Equations for E and / 
By Eq (7) dE = dQ- PdV 

Hence by Eq (20) dE = K v dT + UV - PdV 

= K v dT +(l-P) dV. 
In lieatmg at constant volume dV = 0, hence 

dE ] =K (84). 

,7/71 / "- v ' ' 

dl J v 



270 THERMODYNAMICS 

In isotheimal expansion dT = 0, hence, using Eq (21), 

dT ' P 



We may theiefore write 



-P 



Again, by Eq. (9) dl = dQ + VdP. 
Hence by Eq (24) dl --= K v dT + I'dP -I- VdP 
= K v dT + (l r + V] dP. 
In heating at constant picssuie dP = 0, hence 



dT 



-K 

~ J> 



In isotheimal compiession (IT = 0, hence, using Eq (25), 

( i T i_ T/ _ v T I \ 
77, I = I + r ~ t ' JL I I 

dPJ T \drjj, 



We may theiefoie 

dl = K v dT + I T 



T 



10) 



182 The Joule-Thomson Effect In <i Ihiolllmg pioccss 
dl = (Ait 72), hence, fioin Eq (89), 
'dT, __ 1 
SIP ] i~ K 

This is the "cooling cllect" in Ihc .Jouk -Thomson poioiis plug 
expeiiment ol Ail 19, the cooling cllc( I which Ihc \\oilving Mind 
of aiefiigcrating machine undcigocs in passing Ihc ( \p,insiou-\ ,il\ c 
(Art 110), the cooling cflcct used cuinul<ili\ t lv l)\ r Ijndc foi I In 
liquefaction ol gases (Ail 123) It cxpi esses Ihc fall ol Iciupci.ihuc 
per unit fall of ])rcssure when any fhiidsuKcisa lluolllingopt i.ihoii, 
during which it icccivcs no heat fioni oulsidc. 

From Eq. (10) it follows Hut the cooling cl'lccl vanislu s wlu n 



This occius in any ideal "peifcct" gas under all conditions, lh.il 
is to say in a. gas which exactly .satisfies the equal ion /*/' RT 
I3ut it also occurs in leal gases under particular condihons of 
tempeiaturc and pressnic A gas tested lor Ihc Joule-Thomson 
effect at moderate pressinc, and at vaiious tempera hues, will he 
found to become waimci instead of colder on passing Ihc plug- if 



ir J GENERAL TIIERMODYNAMIC RELATIONS 27? 

he temperature exceeds a ccitain value At that temperature, 
/Inch is called the temperature of mversron of the Joule-Thomson 
ffect, throttling produces no change of tcmpciatuic Above the 
empcraturc of inversion the effect of passing the plug is to heat the 

fdV\ V 

as , is then less I Iran and the expression for Ihc ' cooling- 
\dJ. lp J 

fleet" rs negative. Ik-low the temperature of inversion the cooling 
fleet is posiluc. Tlu temperalme of inversion depends to some 
xlcnt on llu 1 piessuie, m any one gas It chU'eis widely rn difleicnt 
ases In an, oxygen, carbonic acid, steam and most othu gases 

is so high thai the normal effecl of Ihiollling is to make the gas 
older, mhvdrogeii,on Iheolhei hand, the normal effected' I hrot I ling 
, to make lire gas warmci, lor lire lempeuilure of mvcision is 
\ce})tionally low, about - <S() U C 1 * In lire Lmde process it is 
ssenlial lh.it I he gas to be liquefied should enlei lire app.ualus 
I a lempeialme below its temperalme ol nneision the pioecss 
an be applied lo h^diogcn only In cooling I he gas beforehand to 

suitably low lempeialme 

Taking Eqs (JJS) anel (10) logelhei we ha\e 



'his pioelucl, A'j, (-7 ) , is Ihe qiianliU ol heal lhal \\onlel just 
\dl / f 

ulliee lo neiihah/e Ihc Joule-Thomson cooling cllee-l ])ei uml 
lop in piessuie, if il \\cie supplied lo the 1 lluiel m llu> piocess ol 
In ol limn 1 1 ma\ coin e me nll\ be ie pie scnle'd 1>\ Ihe single s\ m be >1 
II me-asiues Ihe 1 cooling clleel, pel uml elmp in piessuie l>\ 
luollling, as ,i (juanliU ol heal (e \piessetl m woik umls), \\lnle 

...I measmes lhal e-llecl as <i change m le mpe-ialme I 
dr i 

II lollo\\s lhal il Ihe i.mge lluough which Hie piessme (alls in 

llnedlhng piocess is liom /' , lo P tl , Ihe ulwle- (jnanlity e)f heal 

lial A\e)iilel ha\e lo be supplied lo neulrali/e UK- coeilmg effect is 

l^ 1 p di> = I*' 1 kf^,) - ?/ | (lp > 

s was si a I eel m a footnote lo Ail. 121 | 

* Tluu wan found l>y Ols/owwla foi a juosmuo-diop fuiin 117 atinosplioios to 

aLnioHpheno 

h Jn Callorulars Hit-am Tables UK- quantity here called p la tabulated Cor Htoain 
uloi the heading 'VS'6 1 " (Hoc Ait, 103) 

\ Cf K BucUiuglmm, Bulletin of Ihc. Bureau of Standards ( Washington), vol (5, 
)09, p J25 



278 THERMODYNAMICS [en. 

Since I = E + PV we may write Eq. (41) in the loim 



This is instructive as showing the analysis of the Joule-Thom- 
son. effect into two paits When an impeifcct gas 01 vapour is 
throttled, that pait of the effect which is measured by the first 
teim anses from the fact that the internal energy is not constant 
at any one tempeiature but depends to some extent on I he picssuie 
In othei woids, the first teim is due to departure fiom Joule's 
Law Theie is in gencial an additional pai t of the cflect, mcasuicd 
by the second term It is due to depailure fiom Boyle's Law, 
accoiding to which PV should be constant (bi constant tempeia- 
ture A gas may confoim to Boyle's Law at a paihculai tcmpcia- 
ture and still be impeifcct in thai case il will show a cooling cffccl 
due to the fust term alone It is only when bolh lei ins \aiush 
that the gas is peifect 

Expenments which \\ill be mentioned in the next ehaptei sho\\ 

fd(pr)\ 

that in an impeifect gas the leim --- I may l)f (ilhci 

V (il ' r 

negative or positive accoiding to the conditions of piessmc and 
temperature (Ait. 197) Hence that pait of I lie Joule-Thomson 
effect vi Inch is due to deviation fiom Bo\le\ Law \\ill mulct some 
conditions assist, and undci olhci conditions oppose lli.il paitol 
the effect which is due to deviation tiom Joule's Law The lallu 
pait is always a cooling effect, the foimci may be eillx i <i cooling 
01 a heating effect Al the tcmpcialmo of imeision UK l\\o[),nls 
cancel one anothei 

It may help the student to undcisland Eq (M //) if we put llic 
physical mteipiclation of that equation in aiiollici AV,I\ Suppose 1 
unit quantity of any fluid to undcigoinul diop ol pu'ssiue in passing 
a poious plug 01 olhci tluotthng device \\ r i i ma\ then pul dl* =- -- 1 
Suppose also a quanlilv of heat p to be supplied toil fiom outside 
which i list prevents any change of Lempeialme Then Kq (41 a) 
takes the foim p = dE + d (PV), 

which is equivalent to saying that in the complelc piocess, 

Heat supplied = Incicnsc of internal Kncigy -j. Woik done by 
the fluid 

Here d (PV) is the net amount of work done by the fluid, because 
it is the excess of P 2 V*, which is the woik done by the fluid as it 



vrr] GENERAL TIIERMODYNAMIC RELATIONS 279 

leaves the apparatus, over PJ 7 ^ , which is the woik spent upon the 
fluid as it enters the appaiatus. 

183 Unresisted Expansion ___ In. the Joule-Thomson poious 
ping expciinicnt the fluid, in expanding from a region of constant 
high piessure to a region of constant lowei piessure, does some work 
on things external to itself, the ncl amount oC Avhich is 

P V P V 

1 2' > * i' i 

This quantity is not zcio except in special cases. 

But in the oiiginal Joule experiment with two closed vessels 
(Ait 10) Llic fluid did no woik on anvthiug external to itself 
The expansion thcie may thciefoic be described as stuctlv un- 
)esif>tt'd This distinction between it and Hie Joule-Thomson mode 
of expansion is unpoitant 

Imagine the Iwo closed vessels of the Joule cxpeiimcnt to be 
completely impcivious lo heat, so that no heat passes oul of, 01 into, 
Hie fluid as a whole dining Ihc piocess Imagine also lhal heal ma^ 
pass fice.lv fiom Ihc fluid in one vessel to the lluid in the othei 
tlnough the o])enmg between them, so that aflei expansion T 
becomes Ihc same in both as \vcll as P Under Ihcse conditions 
I he internal encigy E of Hie lluid as a \\holc is not allticd by the 
expansion, loi no heal is taken in or given out, and no vunlc is done 
This is line ol any Hind The chaiac kiishe, theiefoie, of such 
expansion is that 1<] is unchanged, |usl as the chaiacU nslie of the 
Joule-Thomson expansion is thai / ib unchanged 

In the uniesisled Joule expansion e.ich \essel nia\ ol couisc be 
ol any si/e Think of I he second \essel, mlo \\hich Ihc fluid e x- 
pands, as eonsisling of a gioup ol \e'i\ small chambtis which <uc 
suc i eessi\ ely opened, so I hat Ihe \olume ol Ihc lluid meieases \i\ 
sleps, each d]' \Ve slill suppose UK lempcialme ol the lluid lo 
allain e([iiilibi mm al each slep, and net heal to come in fie>m oul- 
side Then (01 each sle|) (IE = With infinitesimal steps the 
piocess becomes continuous The cooling c fleet in this imagmaiy 
process is not identical with the cooling eflccl in Ihe Joule-Thomson 

experiment In this pioccsb it is ( .) , namely the late 1 at 

\(lr ; i? 

which the lempeiatuic falls \\ilh mciease of volume, under Ihe 
condition that E is constant 
By Eq (30), wntmg (IE = 0, 



280 THERMODYNAMICS [en 

and this, along with Eq (35) gives 

K f dT ] -T( C!P \ p-( dE \ 

~ R '- 1 " 



Eq (4-2) expiesscs the coohnq eflecl in this imas>imuy pioccss 
as a fall of temperatmc, pci unit incieasc of volume, Eq (43) 
expresses it as a quantity oi heat, pci unit incieasc of volume, 
namely the quantity that would have to be supplied fiom outside 
to nentialize the change of tempciatuic caused by the expansion 
We may call this quantity of heat <r 

Hence in nnicsisted expansion fiom any volume V L to any 
volume F ' B , unclei adiathermal conditions (Joule's expansion with 
\essels made peifectly unpeivious lo heat), I he \\holc quantity 
of heat that would have to be supplied to neuliah/e the cooling 
effect is, foi anj fluid, 



/ 
A fiuthei intciestmg lelaLion follows. By Eq (28), we had 



K K = T - 

/V 11 ^ I ~ J- \ ,/,/T , 

But by Eq (35), 



AUo.bj.EnW. 

On substituting these values, Eq (2) lakes the ue\v lot in 

A' -A' _(Z + -M F +/>) nn 

JV n ^ D ~~ T ~ 

This, hkc all the iclations i\en in the pic-sent eha])lcv, is line of 
any fluid. We shall letuin to it latei in connection with unpeifect 
gases (Ait 19-t) 

184 Slopes of Lines in the /c/>, T(f>, and //* charts, for any 
Fluid. The slope of=any constant-pi essurc line m the 7c/> chtnL is 
equal to the ahsolutc tempciatuic, foi, by Eq. (1(5), 



It follows that all constant-piessmc lines in that chait have the 
same slope at points whcie they cross airy one line of constant 
tempeiaturc, 



vnj GENERAL TIIERMODYNAMIC RELATIONS 281 

To find an expiession for ( - ) , which is the slope of a constant- 

\f/0/2' 

tcmpciature line in the 70 chait, we shall piocccd by a process 
of substitution \\hich may be followed in finding othei paiLial 
differential coefficients. It will serve as an example of a gcncial 
method 

Starting with Eq (9) 

dl = Td<f> -l- J'clP, 

we shall eliminate dP by substituting foi it an cxpicssion in tcims 
of dfi and dT, got by applying Ihe gcncial iclaLiou of Eq (I), 
namely, 



This subslitution gives 

''-[ 

Ileucc, wilting dT = 0, 

'(II 



i v /, ,N 

SM.CC, b, JXI (1 I), T 

Sinulailv, to dud an t\[)icssioii loi f ) , which is I he slope of 

d(p i 

<i conslanl-\()luni( lint in llu f<p eh, ul, we stall horn the same 
(qualion loi f//, bill eliininale <U* bv substituting an t'\pu ssion I'oi 
il in Icims of dcp and dJ'", namely 



This siibslilut ion ives 

*' 



til -- T -|- 

J Fence, willing dl r = 0, 

flf\ 

= T -i- V 



fl<f> / 1 

asT - J '(f^) ii ( I0fl ) 

Ocj (12), 



282 THERMODYNAMICS fen 

Turning next to the T<j> chait, the slope of a constant- volume 
line is given by Eq. (29), dT\ T 



arid the slope of a constant-pi essiue line by Eq (30), 

fdT\ T 
(df) P ~~K~ f 

To find the slope of a line of constant total heat ( . . ) we may 
1 \drf>J[ J 

again apply the method of substitution. Stai Lino with the equation 

Td<j> = dl - VdP, 
substitute for dP an expiession in dT and. ell (Eq (I)), 

dP - f flP } dT + ( dP ] dl 
(IF ~ + ai 



d 



ml ni i 1 -. ~rr [til \ lr __ /uZ \ ,. 

This gives 7rf0 =\l-V ( Tli ] J dl - J ( dT )dT, 



fioni which, writing dl = 0, 
dT\ 



But by Eq (40), 
Also, since 



(dV\ (dV\ (dl\ dl 

\~FTi = 7r T,r, il 

\d1Jp \(IlJp\flTjp 



^/7V/> 

we ma> put this result m the Loim 
'dT\ T r l 



In the IP chait the slope of an achabatic, or lino of constant 
entiopj, is given by Eq (17), 



fiom which it follows that all adiabatics luive the same slope at 
points wheie they cioss any one line of constant volume 

The slope of a line of constant tcmpeuiLme is given by Eq (38), 



VIT] GENERAL TIIERMODYNAMIC RELATIONS 283 

To find cxpiessions foi the slope of a line of constant volume 
(dl\ 
I ,/p ) > we may pioceed thus- 

\llL ] y 

dl = dE + d (PV) = dE + VdP + PdV. 

- 



(w) . 
By Eq. (Jil) tins may be wnLlcn, 

, -"-*(). 

rwoothei cxpiessions which uc sometimes useful m,iv coin cm- 
ently l)e o.ven heio, one To. t^} ; ,.ul one foi 



dV ' r 



--"'-'i 



185 Application to a Mixture of Liquid and Vapour 
in Equilibrium Clapeyron's Equation Change of Phase 
Eqiialion (30) is applicable uol onlv lo liomoociieoiis fluids, buL Lo 
a mivlme of l\vo phases of Ihe same snbslance, in ((]iiilibnum wilh 
each olhei and Iheiefore bolh aL I he s.nuc picssuic and Lhc s.unc 
lempd.iline 7 and V uc Ihen Lo be teckoned foi Ihe mjxUuc 
as a whole Say loi mslancc IhaL Lhc snbslanec is a mixluic, paiL 
h(]iiid and pail saLmated va|)onr Suppose Lhc ])iopoiLion of 
liquid Lo vapom Lo be changed by vapormno some of Lhc liquid paiL 
at constanL pressure, and Lhciefoic also at consUml Lcmpcratme. 

During) lhal piocess f-^\ is constant, for the volume of Lhc 



284 THERMODYNAMICS [ CII . 

mixture as a whole me] cases in proportion to the heat taken in. 



Instead of (- r m equation (50) we may thciefoie wnlc 
J p 



I - I w 

( i_ /-\i 



_ _ . -- 

V V V V ' 

' ~ f w ' -. ' iw 

whcie the suffixes s and re iclatc to the two stales, when all is vapom 
and all is liquid respectively Fuilhci, the condition that $ is con- 

dP 
stant may be chopped m wilting the coelficienl wliic-h is 

no longer a pmhal diffeiential coefficient. Since the vapour piesent 

dl* 

in the mixtiuc is ah\avs satmatcd, P is a fund urn of T only, 

is simply the rate at which the piessme of sal million uses with 
the bempeiatmc While the mixLme is vapoii/,mg 01 condensing 
under vanable piessme it makes no diffeience in llu iclalum of 
P to T whcthci the pioecss is conducted \\ilh </> = constant, ot 
with V ~ constant, 01 m any othci way dining Uul pioccss 

fdP\ fdP\ . (IP ,. . 

( -iTiJ 01 7>.J IS tlw same as ., Ilcnct wlien applied lo ,m 

\flTJ ^ \dT)i dl ' ' 

equihbiinm mixtiuc of liquid and \apom, 01 o( any I wo [Aliases 
Eq (50) may be mitten in the foiin 



01 V - V = LdT 

Ul ' s ' w 'f ( Hi 

This is Clapeyion's Equation, which was ainved <il m \il <)S in 
anothci -\\ay 

The same lesult may be got from Eq (21) 



,dV) T 

During vaponzation at constant tempcralme ( ) is eonstanl 

\al /'/' 

and its value is -- . Hence, chopping (lu snllix /' loi the 

' s ~~ V w 

reason just given, we have as before 

FS _ FW = ^ 

This icsult may be extended lo any reversible change of phase 
which a substance undeigocs at constant pressure Dining any 
such change the two phases of the substance aie in eqmhbuum 
with one another and the tcmpeiatme is constant. Wilting A for 



vi rl GENERAL TIIERMODYNAMIC RELATIONS 285 

I ho heat I, ikon in dm ing llic change of phase, and ("'and V" foi Ihc 
volumes of Ihc fusl and second phases icspcclivety (as in Ait 99), 

we have \ (IT 

V" _ //' - Cn/7 1 ) 

' ' TC/P ' ^ '' 

Sunilaily, the cxpussion for ( ) in Eq. ( 1<7 b), namely 

\9// 

/v/r\ = 2L_ TZ ( (W \ 

ttyJrK, V \dl)i.' 



may 1)C adapted lo a mixture of liquid and \apoui in equilibimm, 
during the change of phase which occuis in \apoi i/ahon at con- 
stanL prcssuie (and temperature). In llus pioccss K,, is infinite, 
foi heat is taken in without use of tcnvpciatmc, and alse> 

dV 



The equation Lhcicfoic takes the foim ' 

idT\ T- V I' 

f - '" 



(52) 
V L ^ ! 

This applies al an\ sLage in Ihc process of \apoi i/almn, V being 
the volume of Ihc nuxhiic <il lhal slagc, nameh q]\ I- (1 q] f',,,, 
\\Iioic q is the fiaelion I hat has been vapoii/cd ( Vil 71) It gi\e-, 
Ihc slope ol a line ol constant lot.il beat in UK \vc I icgion (Ihc 
legion willun Hie boundai\ eiirxi) ol the 7V/> ch.ul 

A still moie diiecl means ol getting Clapu loifs E<pialion is lo 
use the function (1, \\liich is 7V/> - / 01 (, 

HvEq (10) 



III an\ change ol phase \\ Inch oeeuis al conslanl lempeiatiue and 
const, ml picssmc, such as the con\ eision ol walci into steam at 
conslanl pussiue, (IT and <ll' ate bolh /eio Ilenec in such a 
change U is conslanl, as was pointed oul m Ail 90, whuc Ihis 
piopeih of (i was luincd lo accoiml 

C'ompaic now the slate ol anv sub.slance at the beginning and 
end o[ a change of phase, dm ing which G 1 is const, mt Use the 
sullix 10 foi Ihe (irsl stale (sav uatei), tmtl Ihc sullix ,s- lor I he second 

state (sav sleam) ( < si **>* 

N ljr f - t^Mi V JJ ;> 

UUr, = (10 w , 

<^ S 6/T - V a dP == <f> w dT - r w dP 

* Uflod by Jonkm and Pyo (Phil. Trans A, vol r>3i, p. 300) ni collecting the Tfj> 
chart foi carbonic acid 



286 THERMODYNAMICS [cir. 

dT 
Therefoie F s - V w = (<f, 8 - <,) ^ . 

But cf} s <f) lo 7 = l . Hence this again give;, Clapcyion's Equation, 

v -v = LdT 

s w TdP 

1 86 Compressibility and Elasticity of a Fluid. Lcl a fluid 
be subjected to an mciease of pressure dP, with the lesiilt lhal Hie 

(.IV 
volume is icduced fiom V to V dV Then measuiob the 

volume strain, and the ratio of this stiain to dP mcasmcs the com- 
pressibility. 

The reciprocal of the compiessibihty 01 Vi y^J measiucs what 

\Ur / 

is called the elasticity of the fluid Its value will obviously depend 
on the cncumstances imdei which the compiession lakes place. 
We may for instance keep the tempeiatme constant dining ihc 
compiession. In that case the expression ioi the elasticity becomes 

- V ( 7rr I This is called the isotheimal elasticity oi a ilmd. and 
\dV) T J 

will be denoted heie by e t Oi we may prc\ r enl any hcaL fiom 
leaving or enteimg the fluid dining the compicssion In that case 

'dP\ 

the expiession becomes -T (-,_.) . This, which is called Hie 

\dVJ {lt 

adiabatic elasticity of a mud, will be denoted hcie by c lt , \V'e ha\ e 
accordmglj' the two elasticities 



T-T 

Hence 

e * = 

d 



byEq (33). That is to say, the ratio of the adiabatic to the isotlu r- 
mal elasticity is equal to y, the ratio of the specific heats Since K 
is greatei than K v (Ait. ISO) e (j) is gieater than e t 

187 Collected Results. All the foiegomg relations arc true 
of any fluid Befoie pioceedmg to apply them (in the next chapter) 



'iij GENERAL THERMODYNAMIC RELATIONS 287 

o particulai fluids, it will be useful to collect them here for con- 
Tmcnce of reibiencc 

dE = Td<j> - PdV . (8), 

I^E + PV, dl = Td<f> + VdP (9), 

^ = I~T(/>, d = VdP - <(>dT (10), 

/; = E - T<[> , dijs = - PdV - frlT 
'dT\ 



dr)*~ l ' u/ (12) ' 



dT 
dl\ 



(JV } - - f" r } di] 

flTJr [ '>> ( )} 

^)r <"-) (15) ' 



(ir '"' (10), 



dPj T (1?) ' 



(23), 
(20), 
( dK *\ = _ T ( d ^\ , . (27) 

V ,1T> I \,1T2I ' \^' J> 

\ tlJL ) ' \CIJ J p 



288 



HERMODYNAMICS 


[en 


s '- r &}, 


(29), 


*. -($ 


(30), 


!TW),r ~(df) r 


(3D, 


K D fdT\ (dV\ 

T (dP)i~ (cirjp 


(32), 


_ K v _ fdV\ fdP\ 


(33), 


/dP> fdP\ 

\w)r 7 \dv) T 


(33 a), 


fdE\ 

\dr} v ~ K " 


(31), 



P = K 1> 



dV) 



_ 

dPJ T \ dP T 



dTl 



(35), 



dP 



dl = K p dT+ \V-T(~\ dP . (30), 



dp 



dPj T 
/ = -l^J_-l ,-'} ( 4] ) J 



.. - (42), 



ra] GENERAL TIIERMODYNAMIC RELATIONS 280 



(1,5) 

P ( >' 

f// 



, rr , Tr 

= T + V =T -V (10 , 



= _ = _ 

' ' A-, v\di) L y 1} ' 



MS) 

\>"h 



In a icvfisiblc change ol phase at consLmL j)icssiuc 

r -r - LllT 

' s ' 10 rji I p 



id (; = C;,,,, or 7V/j, - /, = Trf> w ~ /, (5,3) 

Tlit isolhcinial <iiul adujjjalic 



(55), 
(56) 



II T 



CHAPTEE VUI 

APPLICATIONS TO PARTICULAR FLUIDS 

188 Characteristic Equation The genual Ihermodynamic 
iclations consideied in Chaplci VII can bo applied lo dclcimnic I he 
piopeities of a paiticulai (hud wlicn an equation connecting one 
of its pioperlies with t\\o othcis is known An equation ol Ihis 
kind is, called Lhe "Chaiactenstic Equal ion" or "Equal ion ol 
State" loi the given fluid. It is based upon expu unc uLil know- 
ledge of how the numerical \aluts ol sonic out pioptih, such as 
the volume, depend upon those ol hvo olhci pioptilies, sueh <is 
the piesbiue and the tempeiatuic, Ihcse two being used ,is inde- 
pendent \anablcs loi specif} ing Ihc stale The 1 most usual loim ol 
characteristic equation is one connecting l r with /' and T Sueh 
an equation, when it can be established, is ol lund.inu ulal iiupoi- 
tance m the calculation of oLhei piopeilies Jiul lakcu by ilscll il 
cloeb not allow all the thcimodynanne quaulilies lo be <h leinimed 
foi that puiposc it must be supple menletl by dal.i ugaiduig Ihe 
specific heat, 01 (what conies to the same Hung) l>\ dal.i ,is lo Hit 
lelation of the internal energy lo the temptiahne 

189 Characteristic Equation of a Perfect Gas The simplest 
case to considei is that of an ideal g.is coid'oniiing exacllv lo llu 
equation j>y / t >y 

where R is a constant and T is Ihe absolute lempeialuie on Ihe 
theimodynamif scale We discussed some of Ihe piopei lies oi sueh 
agas m Chaptei I, but iL will be mstiuelive now, as a firs I example 
of the method, to show IIOAV certain results which weie obtained 
there follow dncclly when this ehaiaelenshe equation is mlcr- 
preted by applying to it some of Ihc geneial ieli lions oi Art 187, 
which hold for all fluids 

By diffeientiatmg the characteristic equation ol the ideal gas, 
we have 



ir. vm] APPLICATIONS TO PARTICULAR FLUIDS 291 

[ence m such a >as, 

P 



- 
dT) v ~ V~T' \dT)p~P~T } 

d z P\ _ 
" 



y Eqs. (23) mid (27) of Chap. VII, m any fluid, 

dK,\ _ T (d*P\ fdK,\ r 

dr ) T - L (dT*) y dlld I dP ] r = ~ J 
ence in the ideal >as, 



[uis it follows fiom the chaiacteristic equation that both K ' and 

3, aie eonslanl at any one tempeiatme, m olhei \\oids they aie 

dependent of the piessmc They mav ho^e\ ei \aiy vullitcmpeia- 

le Ihe chaiaeLeiishc equation yi\es no mfoimalion on that 

)int 

By Eq (28) ol Chap VII, m any llmd. 



./V. JJ -tY. y J. 

Liicc m the ideal <>as, 



z r 



This apices willi Yit 20 The lacloi --/ is onultcd because quan- 
les ol heal aie hrie expiesscd in \voik mills (-Vil 179) 
Hy Kq (10), Chap VII, in any llmd Ihe cooling effect in the 
ule-Thoinson poious |)lu^ evpeiunenl js 

7' ('"" 

V' 

V 

the ideal ya.s ( "' ) = -, , hence the quantity m square brackets 



-pdv. 



and theic is no eoolmo cffccl. 
By Eq (30), Chap. VII, m any fluid, 



the ideal gas T [ Tr =, ) = P, hence 
\aJL j Y 

dE = K v dT, 



292 



THERMODYNAMICS |c" 



and since K is independent of the pressuie it follows that the 
internal eneigy of the ideal gas depends upon I he tempci aline 
alone. 

By Eq. (39), Chap. VII, in any Quid, 



In the ideal gas T = V > hcuce 






and since K v is independent oi Ihc picssuie it iollows that the 
total heat of the ideal gas also depends upon I he Unipeiatnrc alone 
These icsnlts show that a gas which conlbims exactly to the 
chaiacteiisiic equation PV = RT (T bcino Ihe lempcialine on the 
theimodyiicinnc scale) conforms exactly bolh lo Boyle's Law (PV 
constant foi any one tempeiatuie) and lo Jonle\ Law (E a function 
of the tempeiatuie alone). It is therefoic ''pi-ilccl " in Ihe sense of 

Art. 19. 

When the equation PV = RT was inliodueed in Ail IS Ihc 
s} mbol T denoted tempeiatuie on Ihe scale oL Ihe gas Iheimomelei, 
that is to say a scale, defined by the. expansion ol Ihe gas ilsilf, and 
the gas was assumed to confoini e\aclly to Bo} le's Law Bnl il it 
also confoims exactly lo Joule's Law, Ihe scale ol Ihc <>as Ihu- 
mometer coincides, wilh Ihe Ihumoeh nainic scale (Ail \->) 

190 Isothermal and Adiabatic Expansion of Ideal Gas 
In the ideal gas, since E depends upon Ihe lempeialme alone , it is 
constant dining isothcuual expansion, and theieioie (he \\oi k done 
by the gas is equal to the heat it lakes in The picssme. \aiies in- 
versely as the \olunie 

By Eq (33 a), Chap VII, foi the adiabalic expansion ol any llincl, 

'dP 



Hence in the ideal gas 



So that in the adiabalie expansion of an ideal gas, 
dP cW_ 

p- + y v - 

If now we make the fuither assumption that y is constant, which 



en] APPLICATIONS TO PARTICULAR FLUIDS 293 

equivalent Lo assuming thai the specific heal docs not vaiy with 
mpciaturc, this gives on integration 

ISL ^ J + 7 ly<. 1' consLanb, 
PV = cons Ian L, 

hich is Lhc adiabatic equation of a peifecl gas with constant 
>ecific heat, ai lived at otherwise in Ait 25. 

191. Entropy, Energy, and Total Heat of Ideal Gas By 
qs (8) and (9), Chap. VII, in any fluid, 

,, _dE + PdV _ dl - VdP 
fl( P ~ 2' ~ T 
i the ideal gas 

dE = K,,dT, dl = K v d r l\ 

, P I? 

id since ^ = , 

. . rr dT ,, dr 

(IJ> - A,, I- A' 



P 

cncc \f\\e aqain assume thai Ihc specific heiit docs not vai^ \vilh 
c tempeiatuic, 

E - K,,T | cons I an I, 

/ --- A ,,T -| constanl, 

r/ = K l loo, T + li loq t V -|- conh-lanl 
A',, loq, 7 1 /? loo f 7-* + eonsl.iul 

The values of the constants depend on \\hat inilial stntc js chosen 
I he slaitmo-poinl of the leekomnq. H is only changes in E, /, 
id r/> I hat can lie determined by Ihese fommlas. 

192. Ratio of Specific Heats Method of inferring y in Gases 

om the Observed Velocity of Sound We saw (Ait 18G) lhat 

any fluid the latio y of the two specific heats, Kj,/K v , is equal 

the lalio of the adiabalic elasticity c tj> lo the isothcimal elaslicity 

Also that /f jn-, 

"<--aH, 

cncc in a gas foi which PV RT, 

c t - V =- P, and Cll> - yP 



294 THERMODYNAMICS ICIT 

This i elation has been used as a means of finding y cxpen- 
inentally m an and othei gases which ab ordinal y temperotuics 
and pressuies very neaily conform to the equation PV = RT 
The method is based on Newton's theory of the transmission oi 
waves of sound Newton showed that waves of compicssion and 
dilatation, such as those of sound, tiavel thiough any homogeneous 
fluid with a velocity which may be expiessed a^VeV, wheie V is 
as usual the volume of the fluid pei unit mass (the leciprocal of the 
aveiage density) and e is the elasticity, in kinetic units. It was 
afterwards pointed out by Laplace that in applying this result to 
the passage of sound through an or othei gases e should be taken 
as the adiabatic elasticity e (]l , for the compiessions and dilatations 
follow one another so fast as to leave no time for any substantial 
tiaiisfer of heat fiom the poitions that aie momentarily heated by 
compiession to those that are momentanly cooled by expansion. 
Hence in an under atmospheric conditions, or in any othei neaily 
pei feet gas, sound tiavels at a late equal toVyPV This fact is 
used as a means of cleteiminmg y by measuimg the velocity of 
sound 01 (what comes to the same thing) bv measuimg the wave- 
length in sound of a known pitch 

In an at C and a picssuie of one atmosphcie the \ allies gi\ en 
by \aiious obseiveis for the \elocitv ol sound lange (join 33,000 
to 33,240 centmietics per second ' Under these conditions the 
volume of one giamme of an is 773 1 cubic cms , and P is 
10133 y 10 dynes pei sq cm (Ait 12) Hence, takmo an avei age 
of 33,150 for the velocity, 

33,150 - \/y x 1 0133 - 10'- > 773 1. 
which gn es y = 1 103 

193 Measurement of y by Adiabatic Expansion Method 
of Clement and Desormes. Anothci method of dcteimuung the 
value of y in a gas is bv an expeiimcnt due originally to Clemen I 
and Desoimcs and impioved on bv Gay-Lussac and olheis A 
quantitv of the gas is contained in a large vessel at a picssuie sonu - 
what higher than that of the atmosphcie, and at almosphei ic 
tempeiatme Theie is a pressure-gauge attached, and a lap which 
may be opened to allow some of the gas to escape quieklv On 
opening the tap, the piessure falls suddenly to that of the almo- 
spheie when this happens the tap is at once closed Then the 

* See Raylcigh's Tlieoiy oj ftuiind, vol ir 



mj APPLICATIONS TO PARTICULAR FLUIDS 295 

iicssmc of the gas that icmains m the vessel slowly uses, because 
he tcmpeiahne, which had been reduced by the sudden expansion 
f the gas in the vessel while the tap was open, rises gradually to 
he value which it had at fhst, namely the tempciature of Lhc 
uiroundmg atmosphcie. When this piocess is complete the final 
ircssiuc is noted. Let the oiigmal picssure be P l5 the picssure of 
he atmosphcic P., and the final picssuie P 3 The change fiom 
\ lo PI is appioximalcly adiabaLic on account of its suddenness . 
he change from P 2 to Pj occuis at constant volume Let F 19 V\ 
nd F 3 be the volumes ol the gas pet umt mass, at the Ihicc coiic- 
ponding stages. Then F 2 -= F 3 . We have, in the ndiabatic ex- 
lansion, p T/ y _ p v y 

L i' l r * 2' 2 > 

nd since the initial and final tcmpciatuies arc the same, 



. P, 7'o 

Iciu>c 



2 l 



/allies ol'y me accoidmglv I'ound b\ obscn ing these thue presumes 
Oxpeiiments by Lumnici and Pimgsheim, using llus method in 
n impioved fonn, oi\c- 1 J025 as the \aluc ol y loi noinidl an An 
aihei application of Ihc method by Ronlgcn ga\ e 1 105 ' 

194 Effect of Imperfection of the Gas on the Ratio of 
.pecific Heats II has been aluadv mentioned that in a peilecl 
halomic gas Ihe lalio y, as deduced fiom the moleculai theoiy 
see Appendix II) 5 shoulcl not exceed 1 I In an the lalio, aecouling 
o all the evidence, is, at oidmaiy lempcialuies and piessuie-s, 
hghtlv gicaUi This is due paitly lo the presence- ol about 
me pei cent of (monalomic) aigon, but mainly lo I he facl that 
jr is ,in impei led gas, deviating to a small extent boll) fiom 
ioyle's Law and fiom Joule's Law 

H> K<i (It), Chap VII, in any fluid, 



r r _ 

A j, A ,, - 



y, 



vhere p is the cooling- cflccl m the Joule-Thomson poious plug 
xpcnment (Ait 182), and a is the cooling effect that would be 

* Seo Plosion's Tlicmii of Heal, Chap TV 



296 THERMODYNAMICS [c H 

found m unresisted expansion (Ait 183), without gmn 01 loss of 
heat in either case. In a perfect gas p and a aie both nil, and Ihc 
expression on the light becomes PF/T, as it should Willi air (nndei 
usual conditions) both p and a aie small positive qnan Lilies p was 
measured in the Joule-Thomson expeiiments, and a, though iL has 
not been dnectlv measuiecl, can be mfeiied fiom known cxpcii- 
menbal da ta Hence K^ K is a little gi cater than PV/T, which is 
the value it would have in a perfect gas 

The ratio y is also a little greatei in noimal an than it would be 
in a peifecl gas. In any fluid 

(P + a) (V + p} 
y= 1 + K,T 

In an at ordmaiy tempei atiu cs the impcii'cction inci cases 
(P + cr) (V + p) moie than it increases K v and consequently 
makes y slightly exceed the ideal value 1 4 But at high tcmpcia- 
tures K v is much increased (because the molecules then acqunc 
energy of vibiation) and y is substantially i educed 

195 Relation of the Cooling Effects to the Coefficients 
of Expansion The expressions for p and a given in Eqs (<J ] ) and 
(4<3) of Chap VII may be put in anothei foim Mhich is coin enienl 
in dealing with impeilcct gases 
By these equations, in any fluid, 

' 



Heie ( -j=A may be wntten aV, wheie a is the fiactional mcicase 

\al >p 

of volume pel degiee. on the theimodjnamic scale, when Ihc Hind 
is heated at constant piessure. Measuiecl at C a is the co- 
efficient of expansion at constant piessuic, or what is sometimes 
called the ' L \olume-coelTiciciit." 

Similaily (-7) niay be wutten /3P, wheic j3 is the fiactional 

\(LJ. / f 

increase of pressure per degree on the thcrmodynamic scale, when 
the fluid is heated at constant volume Mcasmcd at C. j3 is A\hal 
is called the "piessmc-cocm'cient." 
Hence at C. 

F + Po = 273 la Q V Q> and P + cr, --= 273 lj8 P , 
the suffix being mtioduced to show that the quantities conceined 
aie all to be taken as at C. 
The results of the Joule-Thomson porous plug experiments may 



inj APPLICATIONS TO PARTICULAR FLUIDS 297 

3e used to calculabe p Thev showed thai with an the cooling 
'ffcct of passing the plug was nearly pioportional to the drop in 
jiessure It was diffeient foi different initial tcmperalmes, bc- 
'oming less when the initial tempeiatuie was raised. With air at 
) C the cooling effecL (according to the foimula in Art. 123) was 
) 275 for a pressuic-chop of one almosphcu in passing the ping 
-lence, using c o.s units, foi air at C. we should have 

'(IT\ _ 275 
J 7 ~l 0133 x 10 

kVe may Lake 7v,, as 0211 caloiy (Ait. 161) equivalent in c c, s 
nuts of woik Lo 211 x 4 1868 x 10 7 . Multiplying the values of 

fdT 
K v and . ) we obtain 



This is in cubic ccnhmelics pci giammc, the dimensions of p 
DC'ing Ihc s<unc' as Ihoso of V, namely 

woik \olume 

pussine > mass mass 

We ma\ appl\ I Ins lesnlt ol Ihc poious plug expeinnenl to 
'.ilcul.iU the eoellicKiil of expansion \\hcn an, at C , is heated 
indtr a conslanl picssinc ol one dtmospheie Ihiounli one degiee 
>f I he Ihcimoch'iiainic scale We had 

- r o I- Po 
11 " 27.5 1 T 

In an aL C. and a piessuie ol one atmospheic, Ihc \ ohimc ol one 
gamine it, 773 5 cub cms Hence under llusc condiLions \\c should 

l] ^ e 7735-1-271, 

"* - 273 1 , 773 5 = 0(W 75 

1'his is shi>lilly laigci than the. mean coclliciciil lhal is lonncl \\hcn 
he expansion ol an at a consLanl piessmc ol one atmosphere is 
ncasuicd over a i<ine of IcmpciaUire Irom (\ lo 100 C 
Again, taking I he iclalion 



lvalue of O-Q can l)e infcricd when thepicssiue-cocHicient is known. 
If j8 foi au- be taken as about 003G71, 7 J + cr n becomes 1 003 lP , 
making CT O - 003 \P 
In a perfect gas both coefficients, and /3 , would be equal to 

~, or 003G617 The scale of the perfect-gas thcimomclei, 

itO 1 



208 



THERMODYNAMICS 



LC-II. 



whethei of the constant-volume 01 constant-piessmc type, svould 
coincide at all points with the thermodynamic scale* 

196. Forms of Isothermals. Diagrams of P and J", and of 
PV and P. Taking any ideal gas, which satisfies the cliai aclenslic 
equation PV = ET, leb us diaw its "isotheinials" on a diagram 
whose cooidmates aie the volume and I he prcssuic The chaiactci - 
istic equation shows that these curves ait leelanguLTi hypeibol.is 




Volume 
Fig 84 Piessuie Volume Isothcimals fm a Poifoct Gas 

(ng. 84), ioi while any tempeiatuie lemains consl.iul P v.uus m- 
veisely as V These isotheimals foi an ideal gas slioiild be t-oni- 
paied with those ioi a liquid and its vnpoui ahead v illush.ilc-d in 
fig 14 (\it. 76). to which we shall lecui prescnlly. 

Anothei kind of isothermal cuive, which Ama,il showed lo he 
useful in dealing with ieal gases, is duuvn hv Liiloiii> as cooulmnlc s 
the pioduct PV and Lhe piessine When this nu Ihod is applied lo 
an ideal gas the isolhcimals are simply hon/oulal slum.])! Jmes 
(fig S5),smceat anytemperatuicPI'is consLanl This is.n'ohx inns 

* Refeience should be made to Callondai's papm "On tlu^ ThnnHxlynnnnu,! 
U l re B tionoftlioGasTheimometer"(P7u/ Hag 3>m 1)OU) ], na a, , o.n.i ,,| ], 
the absolute zero may be deteimmed and mteivals on gas and tlu-iiumlyn a muHi.il.s 
compared, by making me ot the Joulo Tliomson cnolinjr ,floU ami UK- moiiH,,,, (I 
coefficients of expansion. 



vin] APPLICATIONS TO PARTICULAR FLUIDS 299 

tesb of whcthei a gas obeys Boole's Law If it docs, then the iso- 
theimals of PV in i elation to P will be horizontal straight Imos 



200 C 



100C 



CL 



o 

- 1 00 C 



Fig 8.3 Amagaf, Isotlici mala lor a Petted Gas 

wlulhci I he gas also obeys Joule's Law 01 no Any cmvalmc in 
these lines, 01 any deviation 1'ioin Hie hoi izonLal, means adipaituu 
I loin Bo\ le's Law. 

197 Imperfect Gases Amagat's Isothermals of PV and P 
No leal gas confoims slncllv to Boyle's La\\ The cxpeiinienls of 




Pressure 

Fjg 80 Typical Amagat Isothoi mal foi an Impel foot Gas 

Andrews, Amagat and others have shown lhat the depaituic fioiu 
Boyle's Law becomes moic and moic marked as the entical point 



300 



THERMODYNAMICS 



CTI 



is appioached Amagat'.s expenments on the compicssibilily <>!' 
oases, which extended up to veiy high picsMiies, show lhal Avhcn 
a line is diawn to exhibit the lelation of PV to P at u conslaul 
tenipeiature, its geneial form is that illustiatecl in fig 80 InsLcad 
of being a houzontal stiaiglit line, as Boyle's Law Mould require, 




in am 

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 
Fig 87 Ainagal's Isotheiimils fo? Cailnmic A( id 

it consists as a rule of two neaily stiaighl pai Is, A and (', one 1 
sloping down and the othei sloping up, milled !>v a snioolh cum 
Theie it> consequently on each isothernuil a minimum value oi P) r 



at a paiticular pressure Foi picssmcs less than Llns 7.3 j 
is negative, foi greater pressures it i-> positive The particulai 



vin] APPLICATIONS TO PARTICULAR FLUIDS 301 

pressure at which the minimum of PV is found depends on the 
tempera! me With using temperatine the position of the minimum 
point B shifts fiist to bhe right and then to the left, and if the 



46 
44 ^ 

42 ^ 
<a 

- 
40 S 

Q. 

38 

36-.. __ 

34 

32 

30 



"" ----"' Pitsstire in aim 

26 ' ' 

20 40 00 80 100 120 140 160 180 200 220 240 260 280 300 320 

Pig SS A. in fij.ru I 's I^uthoimals fui Nitrogen 



44 ^ 

Q- 

42 _ 



40 -S 



qp 
oo 



28'-- 



Pressure in aim 

26 ' ' 

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 

Fig 89 Amagat's Isothei mnls foi Ilydiugon 

temperatuie is high enough it may leach the axis of zeio piessuie 
and disappear, with the result that the whole isothermal then 
consists of an upwaid-slopmg line like BC. 

The geneial features of these isotheimals will be appaient from 



302 



THERMODYNAMICS 



[en. 



figs. 87, 88 and 89, which aie lepiesentatrve examples ot Amagat's 
curves* The tempeiature for which each i&otheimal is diawn is 



14' 



4-100 



13 



T2| 



11 





' Pressure in atm 

20 40 60 80 100 

Fig 90 Witkowski s Isotheimals foi Au 



120 



maiked on it In fig. 87, which relates to caibomc acid, Lhe tcni- 
peiatmes foi which the selected cmves are diawn aie all above Lhe 
cutical point, but the lowest is not fai fiom it The left-hand 

* E H Amagat, Annalcs de Chmue el de Physique, vol. xxn, 1881 See also 
vol xxrx, 1893 



in] APPLICATIONS TO PARTICULAR FLUIDS 303 

ranch of the curve consequently slopes down veiy fast, if an 
othermal were drawn for the critical tempeiature its dnection at 
ic cutical pressure would become vertical, foi at the cutical point 

- ' ) is infinite. At highei temperatures the left-hand bianch 
dl J ' 

opes down less steeply, and within tlie lange of this diagiam the 
iiiiimum point inoveb to the light, but (as Anther experiments 
loved) at highei temperature-) still it moves to the left This 
cituic is apparent in the curves of fig. 88, which i elate to mtiogen. 
lieie the incasiuements weic 1 made at tempciatuics much moie 
imote from the cutical temperature The dowmvaid sloping 
lanch is shoit, and becomes shoiter Avhen the tempeiature is 
used Finally, witli hydiogcn (fig 89), \\heic the cntical point 
as even moie icniole the minimum has disappeaied, and each 
otheitnal is a hue sloping up along its whole couibc. At any \ci\ 
>w ttmpeicituic, howcvci, an isolhennal foi huliogen would ha\ c 
bianch sloping doumvaids, loi modciate values of P, followed bv 
minimum of PV and then an up'Udid slope ]ust as in olhci gases 
ivpeumcnls on an, b\ \Yitko\\ski" 1 , uhosc cui\es aie lepioduced 
i fig 90, show lh.it lot modciate pressings a PV, P isotheimal loi 
11 slopes upuaids all tlic \\av at 100 C (01 o\ei), but at lou 
'inpeiatmcs such as 100 I il slopes steeply do\\ mvaids lowaids 
minimum \alue ol PV and then uses The locus oi the muu- 
mmofJPr is mdicaled in Jig 90 In a dotted line. These conclusions 
ic in lull aceoid with the usulls that have just been slated 

198 Isothermals on the Pressure-Volume Diagram An- 
UKI melhoel ol e \lnbi I ing I he- elepai tine- of leal gases fiom Bo% r lc\ 
,<iw is lo diavv isolhennal Inus on a chaplain ol the 1} pc ol 
g SI-, I he cooiehnales of ^\l]l( l h aie simplv the 1 piessuie and the 
olume. An example ol such a diagiam \\as dc sen bed in Ait 7(J 
'01 convenie-nee of icJ'eience it is lepioduecd as fig 91. At any 
igh tenipcuiLure an isothermal (such as 6') does not differ \ciy 
bviously fiom <i rectangulai hyperbola, but at lower tempeiatmes 

exhibits a point of inflection as m F or E 

Below the cutical tempeiature each isotheimal consists of 
spaiate paits, namely the pait AB, in which the substance is a 
omogeneous liquid, and the part CD, m which it is a homogeneous 
apour. These are joined by the stiaight line BC which exhibits 
le change of phase from hen.nd to vapour Dining that change the 

* FM. Mag , April, 1896 



301 



THERMODYNAMICS 



en 



substance is not homogeneous; it consists oi a mixtuic of the two 
phases, liquid and vapour The loci of B and of C together constitul e 
the boundary curve, the apex of 
which is the cutical point. 

The isothermal for the critical 
tempeiatuie E, fig. 91, touches 
the boundary curve at the ciiti- 
cal point. Its duection at the 
cutical point is honzontal and 
it has a point of inflection theie , 
consequent!}' at that point 

'dP\ .. , fd*P\ n 
, u 



\dV) 



= and 



T 




Fl S 9l 



199 Continuity of Liquid 
and Gas The essential conti- 
nuity of the liquid and gaseous 
states in any substance will be 
lealized if one thinks of a piocess 
by which the substance may 
actually pass fiom one to the 
othei state without any abrupt change, such as that \\hich 
occuis in the boiling of a liquid Staitmg liom E (fig 92), wlicu- the 
substance is a liquid, we might heat it at constant \olunie to a 
tempeiatuie equal to the cutical tcmpciatuu (01 highei) This 
bungs it to H Then it might expand isothcimalh along I he line 
HI, and then be cooled at constant volume fiom / to (' Al (' il is 
a satin ated vapom Dining each of these steps the substance, has 
lemamed homogeneous, the passage fiom liquid Lo \apom has 
taken place in a continuous mannei it would be impossible to 
point to anv stage of the piocess as the stage ol transition horn one 
phase to the other. It is obvious that any isothermal highei than 
the cutical isothermal E would seive equally well loi the step in 
which the substance expands 

The idea of continuity between the liquid and gaseous states ic- 
ceived a icmaikable development in the speculations of James 
Thomson ^ He suggested that we might think of the ciuvcs AB 
and CD as parts of one continuous curve ABJKLCD (fig 93) The 
paits BJ and LC correspond to real phenomena of the kind icfciied 
to m Ait. 79 For in certain cncumstances the pressuie of a hqiud 
* Proc Roy Soc . Nov. 1871 Collated Papers, pp 270-333 



m] APPLICATIONS TO PARTICULAR FLUIDS 305 




Volume 



lay be i educed below the saturation pressure corresponding to 

letempeiatuie, without vapon/atiun, and a vapour may be com- 

icssed beyond its satu- 

itron pressure without i \E 

Diidcnsatron Points be- 

.vecn B and J, and 

etween C and L, ac- 

jrdmgly represent con- 

itions in which a homo- 

encons fluid may tem- 

oianty exist in meta- 

Lablc stales (compare 

1 1. 1 35). But points bc- 

kveen J and L cannot be 

;ah/ed in a homogcnc- 

us fluid they would be 

r jmp!e tcl\ unstable, foi 

liev would reqiine the 

icssuie and the \ olume 

> increase together lf ~ 

Ft nee the eounecling poilion ol lire curxe is no rnoie than a 

i.Uhenralieal abstraction, bill rl allows a continuous expression 

u /' in rclalion lo V to be 

rlcipicted i'or isolhc-inials 

clow the cnhe.il tcnipeia- 

uit as well as lor isolher- 

lals abovt that leiapcia- 

IIH The stiai^hl hue BC 

L-pie-scnls lire ouhnaiy 

loecss o\ xapon/alion or 

ondcnstilion Ml consUinl 

ressuie II is interesting lo 

otiee that the theorclieal J 

onnceting ciu\f, which we 

ray call the James Thomson Volume 

.rave, must satisfy this , , , ,, , 

Fig <J3 Jainos Tlioinaun a ideal isothermal 
her nrodyna nne condition, 

hat the aiea BJK rs equal to the area KLC Foi we may 
oncciVL the fluid to be taken through a complete cycle, from B 
luough JKL to C, and then back to B by the straight lure CB 
)urmg tins cycle its temperature docs not change, and thereibie, 

T! T 20 




K 




306 THERMODYNAMICS [cu 

by Carnot's pnnciple, the woik done in the cycle as a whole is nil 
Accoidmgly $PdV for the complete cycle must \ainsh. hence 
the positive aiea KLC must be equal to the negative area BJK. 

It follows that when we are able to chaw foi any fluid the theo- 
retical isothermal AJLD, fiom a knowledge of the chaiactenstic 
equation, i\e may go on to deteimme Lhe satmalion picssme 
corresponding to the tempeiatuie foi which the curve is calculated, 
since that is the piossure at which the straight line JSC must be 
drawn to make the aiea BJK equal to the area KLC. 

200 Van der Waals' Characteristic Equation A form of 
chaiactenstic equation, applicable to any fluid, was devised by 
Van dei \VaaIs 81 which appioximately expiesses the i elation of P, 
V and T under all conditions of the fluid though any lange of 
density and tempeiatuie, fiom the state of liquid to thai of gas 01 
vapoui at low piessuics, when the behaviour appioachcs that of a 
pcifect gas Although Van dei Waals' equation cannot be accepted 
as exact it gives icsults which coiiespond in a lemaikable mannei 
with the broad featuies that are exhibited by ical fluids, in all 
possible liquid or gaseous states, and tluo\\s light on the phe- 
nomena of the cntieal condition and on the question of conlinuih 
of state between liquid and gas 

Van dei Waals' equation was based on the kinetic thcoi v of gases 
Nomoie than a rough outline can bcgiv en hcieof Lhe consieleialions 
involved in flaming itj. The kinetic theoiv shows that a gas Avlneh 
consists of colliding molecules will confoim to the ideal equntion 
PV = RT onlv if (1) the size of the molecules is indefinitely small 
cornpaied with the space tia^eised by them between Iheu en- 
counteis, and (2) no appieciable pait of the cncig> ol Ihc gas is due 
to the mutual attiaction of the molecules foi one anolhu Ncithci 
of these conditions holds in a ical gas In a ical gas the \ olume ol 
the molecules is an appieciable pait of the whole \olumc oce-upied 
by the gas, and it is onlyaftci making a deduction Joi il lh.il wchavc 
the volume which can be i educed by applying moic press me Again, 
dining then encounteis the molecules atluicL one nnolhci acioss 
shoit distances so that internal woik is done in sepaiatmg them 
The lesult is that this attraction between the molecules assists the 

* The Continuity of the Liquid and Gaseous States of Mallei , published in Jhitt li in 
1873, Eng Tians in Physical Society's Physical Munam,, \ ol i, pail in, 1H ( )1 

f Students asking to puisue the mattei should consult Joana' Di/nuniu nl Tliny 
of Oases, second edition, Chap VI 



fm] APPLICATIONS TO PARTICULAR FLUIDS 307 

Diessure exerted by the em elope in preventing the gas fiom ex- 
pandin If the fiist of these t\\ o effects stood alone we should have 



where /;, which is called the "co-volume." vepiescnts the deduction 
luc to the volume of the molecules But in consequence of the 
>econd effect we have to add to P a term depending on the attrac- 
1011 between the molecules Taking any miagmaiy plane of 
.eparation between two poitions of the gas, the atti action between 
noleculcs across thai plane will depend on the number of molecules 
,vhich aie at any moment so near as to be exeieising mutual foices 
n othei wouls upon the number of encounteis that occm on the 
.eparatmg plane pci unit of time But thai \\ill depend on the 
>quaic ol' the density, foi if is piopoitiomil to the pioduct of the 
nimbeis of molecules pei unit of \ohime on the tuo sides of the 
)lanc Accoidmgly Van dei Waals takes a/V* as the teim which is 
Lo be added to P lo lepiesent Ihe effect ol the mutual moleculai 
itliaclions lie 1 1 cats n and b as constants foi any particular fluid 
[Iis ehaiactciistie equation llieieloie lakes the foim 

P I 

I 

Numciical values of Ihe constants can be found foi any fluid 
by obseixing expciimenlalK ihc lelations of piessuic, \olnme 
ind tempeialuie in dilleient conditions of the fluid, 01 lhe\ 
nay be mlened fiom olhei ex[)eiimental icsulls Van elei 
Waals' equation is intended to appl} to any homogeneous state, 
gaseous 01 liquid ll docs in fact icpiodiice uilh lemaikable eom- 
>iehc nsi\ encss the clucl |)lunomena ol both stales, and also those 
jf the entical i)omt, I hough in some pai heulais it fails to gu e exact 
[iianhlalix e icsults 
II may help lowaids an appiecialion of Ihe physical meaning of 

he leim if we considei llic isotheimal expansion ol a gas to 

ivhich Van dei Waals' equation ap])lics When any fluid expands 
n any manner the he.il taken in, (IQ, is, by Eq ('21) of Chap VII, 

dQ = K v dT -I- T K\ dl r 
[f the expansion is isolheimal this becomes 

,lf\ rit I "* \ jjr 



202 



308 THERMODYNAMICS [en 

Now in a Van dcr Waals gas 

ET a . 

~ ~ ( a} ' 



f I 1 i 

horn which (df) r =v^l 

Hence in the isotheimal expansion of sneh a gas 



- 2 dV ..... (3). 

But PdV is dW, the exteinal woik done during the expansion. 
Compaung this with the general equation 

dQ. = dW + dE, 

we see that in a Van der Waals gas theie is an increase of internal 
eneigy (dE) dining isotheimal expansion which is equal to ^ dV 
We may legard this, as mteinal u.oil\ done against a cohesi\e foice 
~^ resisting the expansion, independently of the exteinal pi essuie 

P. In a peilect gas there Mould be no change of E in isothcimal 
expansion (Art. 189) 

On assigning vaiious constant \alues to T the Van dei Waals 
equation gi\ es isotheimal ciu\ es which ha\ e all the 1 gcncial chaiac- 
teristjos of those shown in figs 8Gto93 When the substance is in I he 
gaseous condition and at any \crv low picssuic, V is so huge that 
the teims ajV- and b become neghgjbly small the gas then ajjpioxi- 
mates to the idealh peifect state and the equation oi\ es ncail\ the 
same lesiilts as those of the pei feet-gas equation PV RT At 
highei piessuies both of the niodihmg turns become impoilanl 

The equation may be \\ntten thus, as a cubic in V* 

W\ T , y aV ab 



This gives thiee loots, leal 01 miagmaiy, foi F, couespondmg to 
any assigned value of P, on any isotheimal When the tcmpeiatuie 
for which the isotheimal is drawn is highei than the cutical Icm- 
peiatme, only one of the thiee roots is ical that is to say theic is 
only one value of V foi each value of P on any isotheimal above 
the one that passes through the cutical point. For any tempera- 



vm] APPLICATIONS TO PARTICULAR FLUIDS 809 

tuie below the cutical tcmperatme all thiee roots aie leal in the 
mathematical sense. The isotheimal ciuve calculated fiom the 
equation then takes the continuous foim conceived by James 
Thomson, and ilhustiated m fig 93. One of the thiee loots coiie- 
sponds to a point on the cuivc AJ t one to a point on LD, and the 
thud to a point (not icahzable cxpenmentally) on JL. 

Van del Waals 1 equation makes the pioduct PV, foi constant T, 
vaiy in the mannci indicated by Amagat's isotheimals, showing a 
minimum at a pailiculai value of the piessmc that depends on the 
tcmpciatiuc foi which the isolhcimal is chaMn Wiitmy the equa- 
tion in the foim RTV a 
PV = _ 

V-b V" 

and diffcicnliatmg with icspect to P, kctping T constant, we have 
d(PV)\ 

( dP } T 

Smccon any isolhcimal - iszeioal the minimum ol PV, the 

quantity within UK sqiiaie bi.ukels musl \anish al Ihal pomi 
Ilcnce, on any isolhcimal, the minimum ol PI' is lound \\hcn Ihe 
is such Ihat 

V V a (M / __ b\*J>R r r 

This shows that llu \ oluiue at \\hich Ihc minimum ol PI' occms 
on an\ isolhcim.il becomes gicatci as Ihe lompcialine is i.usc'd 
In the pailuulai case \\lun the Lcmpcialuic is so high that Ihc 
minimum occms on the PV axis, \\hcic P is /eio, }' is indelmilely 

laige, 1 then becomes equal lo 1, and T is gi\cn by the 

equation T , ,, Ilcnce in a fluid which satisfies Van dci Waals' 
hit 

equation an Amagat isotheimal foi a Icmpeuiluic, equal to 

would slope upwaids along its whole couise, \\il\\ incieasing /*, 
but an isotherm, U for any tempcialiue IOMCI than this would fiisl 
clip lowaids a minimum of PV and then use 

201 Critical Point according to Van der Waals' Equation 
To find Ihc culical point of a fluid which satisfies Van dci Waals' 
equation, we may mosl conveniently wiitc Ihe equal ion m the foim 

RT a 



P = 



V-b F 2 ' 



310 THERMODYNAMICS [en 

t , . /dP\ - RT 20 , ... 

fromwkch _ + . (6). 



, d 2 P 2RT 

and 



At the critical point 

(1*\ =0 and r: T : ) =0. 

\dV/ T \dV~J T 

Hence, wilting T f P c and V c foi the cutical temperature, pressuie, 
and volume, we should have, in a Van dei Waals fluid, 



This ives 



from which V c = 36 (9). 

It follows from this icsult and fiom (8) aboA e that 

RT C _ 2a^ 
(3b^b) z ~ 27b*' 

fiom which T * = &Sb (10) 

Also, fiom the oiigmal equation, 

P = KT <- a = A " _ JL = a n i \ 

c 3b - b Qb 2 27 ft* Oi 2 27b 2 

Thus if the constants a and b as well as 7? \\cie known for a qns 
which strictly satisfied Van dei Waals' equation, the cubical A olumc, 
tempeiature and piessuie might he calculated 01 coincisclv the 
constants might be mfeiied fiom known values of T it P c and V f 

It follows also that in such a gas the bhice cutical quanlitios 
would be connected by the relation 

P V = RT (1*) 

* c c 8 c \ * " 'a 

which shows how widely the condition of the fluid then diffcis 
fiom that of a peifect gas. 

In applying his equation to caibonic acid, Van dei Waals de- 
duced fiom the expenments of Regnaiilt and of Andunvs tlicsc 
values a = 00874, b = 0023, 7^ = 003085, the unit of pres- 
sure being one atmospheic, and the quantity of gas considered 
being that which occupies unit of volume at a piessurc of one 
atmospheie and a temperature of C With these constants the 
calculated cntical tempeiatiue is 32 C , which agices fairly well wilh 
the value obseived by Amagat, namely 31 8 C. In other particulars 
the agreement is less good, thus the calculated critical piessurc is 



A in] APPLICATIONS TO PARTICULAR FLUIDS 311 

61 2 atmosphcies, whereas Amagat's observed value, is 72 9 atmo- 
spheies. This disci c pa ncy may be due in pait to eiiors of expen- 
meni arising from the picsence of some air m the gas*, but jt is 
found that the Van dci Waals equation fails to icprcsent the 
behaviour of a gas veiy accurately in the neighbourhood of the 
ciitical point 

202. Corresponding States. If we ha\ c two or moie diffeicnt 
fluids to which Van dcr Waals' equation applies, \\ilh diffeicnt 
constants foi each fluid, an impoitant relation between them can 
be established by selecting scales of tempciatuie, picssuie and 
volume such that the critical tempciatmes of tlic diffeicnt fluids 
aic expicsseel by the same number, the ciitical piessuics bv the 
same numbei, and the oiitical volumes by the same numbci 
Isotheimal curves diawn to these scales foi the diffeicnt (Inids will 
then coincide in othei woids a single diagiam will show the ic- 
lation of P to V in all the fluids, when it is icad by icfciencc to 
Ihc appiopiiatc scales Similaily .1 single diagiam will show the 
Vmagat cuixes (PV and P) foi all Any point taken in such a 
diagiam, mtcipietcd on the piopei scale, maiks a definite stale foi 
each fluid, and foi the dilleienl fluids it maiks what ,11 c called 
"coiiespondmq stales " The cnlical points for Ihe difltient fluids 
liuuish m ob\ ions example of coiiespondmg states 

Thus fluids aie said Lo be at coiiesponduig piessuics when 
then piesMiies beai the same lalio to the icspcthxc eiitical 
piessuics lhc\ n ic said to be at eonesponding \oluniesuhen then 
\olumes beat Ihc same talio lo the icsptcliv e enlical \olumes, and 
at concspondmg tempeiatuies wlien then lempeiahnes beai the 
same lalio to the icspectne ciitical lempeiatuus If substances 
con foi m to a ehaiaeleiistic equation of I he Van dei Waals U p< all 
Ihicc quantities P, V, and T, simultaneously ha\ e "coiiespondmg" 
\alucs in the sense hcie defined To put this statement m anothci 
form, let Ihc unit of. tcmpcialmc chosen foi each fluid be its 
(absolute) culical tempciatuie, the mill of volume its ciilical 
volume, and Ihe uml of picssurc ils ciitical piessuic. Then Ihe 
same family of cmves, eithei on Ihc pi essinc- volume diagiam 01 
the Amagat diagiam, will seive to icpicscnt the isothcimals foi 
all fluids that confoun lo a charactciistic equal ion of the Van cler 
Waals fvpc 

Thai this is tiuc of any fluid to which Ihc Van dei Waals 

* Van dm Waals, Plu/iical Mtimaii s, p 4 OS 



312 THERMODYNAMICS [cir 

equation applies will be seen by i educing' the equation lo a moic 
general foim Take .any such fluid, in any given state, and wnle 
its piessure P as p,P e wheie p, is the number by which the picssmc 
is stated when we use the cntical piessuie P c as the uniL of piessuic. 
Similaily foi V wnte v,V c -uheie u, is the numbei by which the 
volume is stated when we use the cntical volume V c as I he unit of 
volume, and for T wntc t,T c svheie t, is the numbei thai cxpi esses 
the (absolute) temperatuie when we use the ciitical tempeiatuie 
T L as unit of temperatuie The quantities p,, z>, and t r arc cnllcd 
the "i educed" pressme, volume and tempeiatme iespcctivel> . 
Then 



T-tT =t - 

~ ' c ' 27 bR 

On substituting these values in Van dei Waals' equation, 



it will be seen that the constants a, b and 7? cancel onl, and I he 
equation becomes / \ 

(/>, + fo) ('', - i) M', 0> 

1 - 



The constants that chaiactciized a paiticulat fluid IM\ <- disap- 
peaied Accoichnglv this "i educed" chaiactcnstic cqntilion, as il 
is called is tine of any substance that satisfies a Van dci Waals 
equation, and consequently the loims ol tlic cmxes roimtcliiin />, , 
v r and /,__aie the same for all such substances 

In othei woids, if we compaie any two such substances, using 
sa\ the tempeiatme and pressiuc as mdepcndcnl vmi.ibles foi the 
pin pose of specifying the state, and choose "coiiesponfliuo " \ahu-s 
of the tempeiatme and piessuie foi the two, then the > ohnnes xvill 
also ha\e "coiiesponding" values 

This is the theoiem of coriespondmg stales, fust enunciated bv 
Van dei \VaaIs IL XNas tested by Amagat and found bv him lo be 
nearly true of a numbei of fluids which he examined lliroui>li a 
wide lanoe of conditions, and it has been shown lo hold nppioxi- 
mately in many substances' 1 The validity of Lhe pimciplo docs 
not depend on the piecise foim of the chaiacleiishc equal ion Van 
dei Waals' equation is by no means the only one lluiL would lead 
to the same conclusion. Any charactenstic equation connecting 
* See S Young, Phil Mag , Fob 1892, also Ins book on Kl 



APPLICATIONS TO PARTICULAR FLUIDS 313 

P, F, and T with no moie than thiee independent constants 
(two adjustable constants in addition to R), and giving a critical 
point, can be biought in like mannei to the foim of a "Deduced 1 ' 
equation in which the constants peculiar to the fluid have chs- 
Hppeaied Hence any such equation seives equally well as a basis 
foi the theoiem of coi responding states 

203 Van der Waals' Equation only Approximate Useful 
as Van dci Waals' equation is in exhibiting bioadly the behaviour 
of a gas even m cxtiemc \anations of state, it cannot be biought 
by any adjustment of the constants into exact agiecment with the 
results of cxpcnmcnt It appeals that the actual piopeities of a 
gas aie too complex to admit of complete statement by the use of 
s'o small a numbci of constanls. The quantities a and b of the 
equation aie not stnctly constant, they aie to some extent func- 
tions of the lempeiatuie, 01 the density, 01 both If constants 
aie selected which fit obsenations of the compicssibihty, the 
equation fails to agice \\ilh measuied A allies of the cutical \olume 
cind cntical picssiue Fmlhci liom Equation (12) ol Ai I 201 uc 
should expect the latio RT\PV to ha\e at the eiilieal point the 
value " 01 2 GOT, whatc\ci be the \alucs ol the constants a mid b 
But the obsenations of Young" 1 show that this lalio is not the 
same in all gases at the entieal point, lhat in most gases it is 
about 3 7, but in some it nun be less than 3 5 and in olheis moie 
than I The iclalions ol the- eiilieal lempcialuic, piessme, and 
\ olumc <uc m laet less simple than is consistent \\ilh the loimula 
of A an clei \\.i.ils the ei ideal pomls m dillen nt ae lu.il Minds aie 
not slnelh "eoi lespondinn" stales, and theie is some dep.iitme 
1 1 om I he IhcoK m ol \i I 20'J 

Vgam, liiUmy Ihc \ r an del Waals equation 

/' _ _ 

and dilfeientialmi> \\ilh respect to T, keeping \ r constant, \vc have 

R 

~ V- b 

which means that when a Van der Waals fluid is heated at constant 
volume the increment ol piessme per degice of use in tcmpciatme 
is constant Hence with such a fluid, Avhelhei liquid or gaseous, a 
thcrmometei of the constant-volume, type would give leadings on 
the thcrmoelynannc scale without ce>iicction In othei woids the 

* Lot til 



314, THERMODYNAMICS [fii 

observed piessure coefficient would be independent of the tem- 
peiatme This ib not true of leal fluids. 

Again, if a substance confoimed strictly to the equation of Van 
dei Waals it would follow that K v , the specific heal a I constant 
volume, would be constant at any one tempciatnie, and would 
theiefore be the same foi the liquid as foi the gas at any tcmpeia tuie 
at winch both states of aggiegation aie possible lor taking Eq. 
(It) and again diffeientiatmg with icspect to T, we have 

-o 



Hence by Eq (28), Chap VII, (/) -0 that is lo say K 

\(.IV / f 

would be constant at any one tempeiatme This, howevci, is no I 
confiimed by meassiuements of the specific heat 

Anothei impoitant paiticular m which Van dci Waals' equation 
fails to give lesults that agiee with those of experiment is in I he 
cooling effect of throttling This effect has been measui ed in vai ions 
gases and 1 ^ apours by expeinnents like the poious-plng cxpcimunls 
of Joule and Thomson. Such expeuments show that in any real o.is 
the effect suffeis an inveision when the initial tempciatnie ol llu 
gas is sufficiently high, that is to say, at high tcmpciatmes the 
effect of tluotthng is to heat the gas instead of cool it The lad 
of this inveision can be deduced fiom Hie Van dei Waals equation ' , 

* To show this we may uv Eq (41 a) of Chap Vff, which o\piessc* tho coolniLf 
effect in an\ fluid as fdE\ 



In a Van cler \VaaIs fluid, by Eq (4) above, 



\ 

r, ,v \ . 

Fllrther = ' and 



by Eq (35), Chap VII Hence we should have 

dE\ RT a dE _ a dV 



Adding the two teiras, the whole cooling eflect in a fluid which olio\H Vim 
b' equation would be 

BTb 



By making T suflkiently large the second teim witlun Iho sqiuuo biaokots 
exceeds the first, wbich means an inversion of the effect When Ihc fluid is a gas at 
low pressure, and Fis consequently very large compaied with l>, tho oonddion foi 
inveision is that RTb = 2a in othei words the inversion toinpciaiuio in a gas at 
very low piessuie would be 2a/Rb 



vni] APPLICATIONS TO PARTICULAR FLUIDS 315 

and to that extent the equation is satisfactory But the amount 
of the cooling effect in such a gas as cai borne acid, when calculated 
fiom the Van der Waals equation (with constants which suit the 
form of the isothermal curves) falls much shoit of the cooling effect 
that is actually obseived, and if the constants of the equation are 
adjusted to make the observed and calculated cooling effects agree, 
then the equation does not accord with the observed figures foi 
compressibihby* 

204 Other Characteristic Equations 1 Clausms, Dieterici 
Enough has been said to show that Van clei Waals' equation cannot 
be biought into exact agieement with the deviations fiom Boyle's 
Law and Joule's Law which arc found in an actual gas. The icason 
has ah cad y been indicated that the "constants" of the equation 
aie not stuctly constant. In particular the attraction between the 
molecules, on which a depends, is piobablv a function of the tcm- 
peiatme, although it is ticated in the equation as independent of the 
temperature Various attempts have been made to mod if v the 
equation so as Lo bung it into closer accoid \\ith the known pio- 
pcitics of gases None of the^e ha^ e been completely successful in 
giMng a foimula which will stand all tests lluoughoiit a \ciy wide 
Kingc of states, though in some icspccls the modified equations 
approximate bcttci to the obserx ed facts 

Clausms | gives a characteristic equation which \\e may \vntc in 

the loini 717- ,/ 

P - _ (10} 

1 ~ v-b Tir + b 1 )* ( h 

wheic a' and b', as well as b and It, aio constants On com- 
paring this with Eq (1 o), it will be seen to differ fiom Van cli'i 
Waals mainly by the presence of T in the dcnommatoi of the 
last teim, which expresses the addition to P that is due to mter- 
inoleciilar attractions C'lausms assumes that these altiactions 
become reduced when the tempeiatine uses, he Iheieby gets an 
equation which, while it gives to the isolhcimals the same gene rat 
form as is given by the equation of Van dcr Waals, agrees bctlei 
with the Joule-Thomson cooling effect When the same method 
of finding the critical quantities is applied to rt, by wilting 



dP\ . 7 /&P\ 
-)= and 7^7-, = 0, 
dV/ T \dF*J T 

* See Callendai, Phil Wag , Jan 1903, pp f>S-()0 
f Plnl Mag , Juno, 1S80 



316 THERMODYNAMICS fen. 

one finds that V. = 36 -I- 26', 



T 

-* /. 



_ , a'R 

* ^ 



216 (b + &') 3 

For caibomc acid Clausius gives his constants the following \ allies 
R = 003688, b = 000843, a' - 2 0935, 6' -= 000977, Ihe unil 
of piessure being again one atmospheie, and the quantity of gas 
considered being that which occupies unit volume at one atmo- 
sphere and C With these constants the calculated cnticnl 
tempeiature is 31 C and the calculated ciitical picssmc is 77 
atmospheres. 

Cldusms diaws a theoietical isothennal cmve of piessure and 
volume for caibomc acid at 13 1 C calculated fiom his foimula 
This cmve, which is leproduced in fig 9-i, shows the foim assumed 
by the James Thomson nave in the Clausius type of chaiactenslic 
equation The hoiuontal stiaight line BC, which exhibits Ihe 
piocess of liquefaction, is so diawn that the ciest and hollow ol the 
waxe shall have equal aicas (Ait 199) this considcuilion dilti- 
nunes its height and therefoie fixes the sanitation j)iessiue The 
dotted poitions of the cmve exhibit imaqmaiy stales, eompnsed 
within the chaiacteiistic equation, uhich sen e lo eslablish 
theoietical continuity between the leal state of homogeneous 
liquid AH and the leal state of homogeneous vapom CD. 

A modified and moic geneial type of: Clausius equation is ob- 
tained bv writing 

p _ 



where J (T) is a function of T such as to diminish with 

perature. In the onginal equation of Clausius, f(T) = . Vnu 

der Waals has suggested that/(7') may be e ' J \ , wlicie c is 2 7 I S3, 
the base of the Napienan loganthms, and T t is I lie enlieal lem- 
peratme. In that case, at the ciitical (empeialun. J (T) would 
become equal to 1 

This foim of characteristic equation A\as aclo[)lcd by Molhc r in 
calculating his tables of the piopcilics of caibomc ,ieid :| 

* Molliei, Zcilsrh/ iflf/ii dtc gc^ainmtf Kalla-lndublm, vol n, ISOH and vol ui, 
1890 



vm] APPLICATIONS TO PARTICULAR FLUIDS 317 

-Still another characteristic equation of the same comprehensive 
kind is that of Dietenci ", who writes 

P (V - b) = RT^ f . . . (18), 

\ / \ / * 

Avhere e is again the number 2*7183, and , b and R are constants. 
Like the otheis, this formula is founded on the kinetic theory, 

80 i i i i i l 



70- 

60- -j 

x"" ~"" __^ 

- o BL / ^"~~^5~ 



50 



' 



40- 1 / 

a I / 

w i / 

S \ i 

30 -Q- \ ' 

\ / 

\s 



20- 



10- 



Volume 

i I i I I I 

91- TlK.oHlii.al lyoUiciinnl of COj aL 13 1 (.' accoidini; to the 
equation of f'lausius 

and like them it reproduces the geneial features of isotheinial 
curves under all conditions and gjyes a cntical point Since it 
has only two constants besides R, the pi maple of coiicspondmg 
states holds good foi the iclation it establishes between P, V and T 

It makes the critical tempei attire ^O^TT^J ^ ie cntical 

* Annahn det Physil, vol, v, p 51, 1901 



318 THERMODYNAMICS [en 

volume V c = 2b, and the cutical piessuie P c . Hence at 

cU 

the critical point the ratio RT/PV becomes equal to \e- 01 3 695, 
a value which is in much bettei agreement with obseived icsnlts 
than \\as the value 2 667 calculated fiom Van dei Waals' equation 
(Ait 201) In icspect also oi the Joule-Thomson cooling effect 
and its inversion 1 Dietenci's equation gives a bcttci agieement 
with expeinnent than does Van dei Waals' 

A moie geneial form of the Dietenei equation is obl.uncd bj 
writing T n instead of 2'in the index teim, thus mtioducmg one more 
adjustable constant - 

(19) 



The ciitical tempeiatme then becomes v / y-- . The nimciple of 

V 4>bR ' l 

coiiespondmg states uould still apply to any gioup of substances 
for which a had the same value, since each substance in the gioup 
would still ha\e only two constants individual to itself. 

205 Callendar's Equation. None of these equations is com- 
pletely successful in lepiesenting the beha\ loui of a lluid in all 
possible states. But toi the piactical puiposc of enabling tables 
to be calculated \\hich will show the piopeities of a lluid Ihioiighoiit 
a limited lange of ranation oj state, it is not impossible to fiamc a 
chaiactenstic equation which, by empuical adjustment of the con- 
stants, can be made to apply \\ith sufficient accuiaey and e\cn 
with gieat accuiaey within that lange, though it m.i\ Kul 
entnely when caintcl beyond the lange. A conspicuous example 
ol this less ambitious type of chaiactenstic equation is one which 
Callenclcii has devised and applied to calculate his tables oi the 
piopeities of steam | It sen es to expiess \ eiy exacth Lhc obscn eel 
piopeities of steam within the limits ol piessuie and tenipcKiluie 
that are usual in steam-engine piactice, but it has no application 
to highei piessuies, and it makes no attempt to lepicsent Ihe 
continuity ot the gaseous and liquid states 

This equation, which Callendai takes as chciiac tens tie, ol tiny 
vapour, satuiatcd 01 supeiheated, piovidcd the piessuie is much 
lowei than the cutical piessuie, is 

RT 

V- p -~c + b (20), 

* See Porter, Phil Mag , Apiil 1906 and June 1910 
1 Callenclai.P/oc Roy Soc vol 07, p 2CG, 1900, PJnl Hag,Jtui IQQ'd, 
Bni Articles Theriuodjiiaiuics" and " Vaporisation " 



vnr] APPLICATIONS TO PAKTICULAR FLUIDS 319 

1RT 

where -^- is, as usual, Lhe ideal volume of a, peii'cct gns b is a 

constant lepiescnting the co-\olumc s as in olhcr chaiactenstic 
oqualions, and c is a term which is not constant but is a function 



of Ibe tcmpciatme Callcndai takes c -, \\hcic C is a constant 

and ii is a number depending on the nahne of Ihc gas. The teim c 
repicsents the c flee I of mtei-moleculai loiccs, but instead of ic- 
gardmg ihcsc loiecs as augmenting the influence of I he cxtcinal 

piessiuc (which Van dci Waals did by adding the leim to P), 

Callendai iej)iescn(s by 6 1 Ihen effect in mincing the volume below 
ils ideal value, in consequence ol the "co-aggiegalion" 01 teni- 
poiaiy mtcilmknig of some of I he molecules dunng then en- 
counters Ik calls c the "co-aggiegahon \olume" and tieats it, ai 
the model, lie piessincs within which he applies Ihc equation, as a 
I'miclion of the kmpei.ihne only This assumption Mould ue>t be 
line undei conditions of high elensity, but (01 a gas 01 salinated 
\,iponr al modeiak piessmes i( gi\cs lesults \\hich ayiee lemaik- 
abl\ \\<ll \\illi lhos( j ol expciiiutnt 

lie (oie pioec'iehn^ lo <ip|>l\ r C'alle ndaTs espial ion il mav be iisclnl 
lo point out its icfaliDii lo lliat of Clausius A\'c max \vnte the 
cqu ilion ol Clausius (Kq (1<>) of Vit 201) in the foim 

JfT f^V -^It) 
1 ^ P ~ PT(V ,-//p" M; 

No\\ ,il l<i\\ 01 mod( i.ili piesMiic's Ihc \olume \\ill b< l.uge, and the 
modilxmg Icinis on tin nyhl \\ill be compaMlM i Ix small \\hen 
/ is laigc MK tlUcl ol the scconel kiiu x\ill not be ninth aikied if 

r it i 

ut. 1 lake , , as appioxmiatcly eniuil to ir , and alse> lake P 

(l r -\-li)- " l V 

in (hat leim as appiovimakly equal lo JfT/f \\hen Ihesc sub- 
ihlulions aie made Hit equation becomes 

JfT _ ' 
~ 1> HT*' 1 ' 

,vlneh we may wnle in Callendai 's form 

., itr c 

~~ P ~ 7'" "'" 

vhcre Ihe more jieneial index it is subslitLiteel I'oi 2 as the index of 
l\ and C is xvrillen for a' /If 
Callendai finds that Ihe best agiecment will) obscived lesults, 



320 THERMODYNAMICS 

especially with obseivations of the Joule-Thomson cooling effect, 
is got by giving to n a value which is not ncccbsauly 2 but may be 
gieater. 01 less than 2 accoulmg to the natuie of the gas ! . Foi 
oxygen or mtiogen. 01 hychogen he takes n to be 1 5 foi caibonic 
acid at low piessine good icsults aic got bv taking it as 2, and foi 
steam, in the calculation ol his tables, he has taken it as 1 , 

Itmiibt 1)e emphasized that Callendar's cejuation applies only lo 
gases and vapouis at low and modciate prcssincs That Ihis is so 
will be obvious when one consideis the foim of Ihe isothermal lines 
which it gives on a diagiam of PV and P We may wnlc it 

Since c is a function of T only, and is there ioie constant along any 
one isotheimal, this gives 



'T 

Hence m a gas which obeys Callcndafs equal ion the isolhcimal 
lines would be stiaight, inclined downwaids, wilh incieasing P il 
cisgicatei than i, and inclined upwaids il b is oiealci Hum c Then. 1 
would be no minimum of PV not change ol inclination along an\ 
isothcimal line The equation theieioie can appl\ onl\ mult i con- 
ditions such that the hues aie substantially sliaii>hl, nanu'K at 
low 01 moduatc picssuies htaitmg liom /' -=- llu lints aie in 
fact neail}. stiaight foi some distance, anil as we saw in \il 1<)7 
thev slope down uhcn the tcmpcialme is low and slope up wlu n 
it is high In an} gas, al a sullu icntly low l( ni]K lahne < isnualti 
than b, and an isolheimal line theie will slopt do\\n \s the l< m- 
pcrature inciease,s ioi which the isolheimal isdia\\n,r be conies l< ss, 

since c = , and a tempciatuu is i cached at which the hue inns 

level (c = b) Foi am Inghei tcmpciatiue Mian tins I In line s)op< s 
ii]) like the lines foi hydiogcn in iig 80 

The tempeiatuie al which the sign oi the slope changts will he 
lelatively low in a gas which, like hydiogcn, has a \eiy low eiihcal 
temperatine, and will be i datively high in a gas like eaibonie acid, 
as, might be mfeiicd horn the pimciple ot coiic'spondmg stale's 

In dealing with steam, the limits within which Callcndai has 
applied his equation aic fiom /eio piessme lo 500 pounds pi r square 
inch 01 3i atmospheiesf- Within this lange il is not |)iobable thai 

* Phil Mat/ , Jan 19US, p 95 

( Theciiticalpioasuiein watci-viipoiu IH aliout 20(1 uLiuortplicirH, oi Hay MIX tunes 
as high as the pleasure up to which Callondai's equation IH lu-UI to apply 



vm] APPLICATIONS TO PARTICULAR FLUIDS 321 

any nnpoitant eiror is mtioducecl by ti eating the isothermals as 
shaight lines on the diagram of PV and P. 

Besides lepiesenting accuiateh/ the behaviour of the substance 
within this lange, when suitable values aie chosen for the con- 
stanls, Callendars equation has the very convenient pioperty that 
diffcicnlial e\picssions deduced i'lom it foi thcvaiious quantities 
E, I, 0, K ]l} K v and so foith, by applying the gencial theimo- 
dynannc relations of Chap VII, lake f'oims such as may be icadily 
intcgiated Ilcncc it enables niimencal values of these quantities 
to be calculated, to any desncd numbci of figuics, which will be 
theimodvnamically consistent with one another 

II would lie possible to fix the constants by icfcience only to 
2xpuimcnts on the compicssibihly of the gas at vaiious tempera- 
tines, if sulfieicnlly accmate data of that kind weic available But 
Callendai picfeis (o (i\ Ihem by lefcu-nce mamlvto obseivcd values 
)i the Joule-Thomson cooling effect Then iclalion to the cooling 
effect will be appaient fiom what follows 

206 Deductions from the Callendar Equation. Taking Hit 
Jdllcndai equation, 



mcc c , 

vc have 

dc nC nc . (Vc n (i 

, and - = - 

dT 7 1 "' 1 7" dT* 

c 1 dc d ic\ 

Vlso, since , } i - , rr dT ( T ) = ' 

)ilfciciiliating the equation with icspect to T, keeping P constant, 
dy\ _ /.' _ dc If nc 
\dfJv~ P dT ~ P "'" T 



hfleicntialing- with icspect to P, keeping T constant, 

low by Eq. (27) of Chap VII, in any fluid 

~dPj^ ~ I (dT^Jp 

2L 



322 THERMODYNAMICS . [cu. 

Hence in a gas 01 vapoui to which Callendai's equation applies. 

(dK,\ _n(n+^)c 
( dP ) T ~ T ( ^> 

Integrating, we have 

n (u+1) cP 
A p = -y + A ( 2( >), 

wheie KJ is the constant of mtegiation It is the limiting value of 
the specific heat K }} when P = 0, at the Icmpcialme T lint since 
any gas in that infinitely laiefied condition may be healed as 
peifect, Callendar assumes that K J} ' may be taken as having the 
same \alue at all tcmpciatmcs to which the equation is applied 

It should be noticed that this mUyiadon is peifouned along 
an ibotheimal line, and that the const, ml ol mlegialion is not 
necebsaiily the same ibi othci tempci.ilmes To deal K ,' as con- 
stant when the tempeiatuie is vaned Iheieloit invokes ,m .issnmp- 
tion \vhich is independent of anvllung m llu equation itself 

Again, by Eq (4-1) of Chap VII -\\c had loi Hit mtasint ol (he 
Joule-Thomson cooling effect in am fluid 

dT\ 



Hence for a gas to which the Callcntlai ttpial ion applies, llu eoohn 
effect is 



RT 

- p V nc - / 

= V H- c - b -! uc - 
= ( + !)< -b 



As was explained m Ail 182, () is (he toll ol Umpeialuu 
pei unit fall of piessure whc>n the gas passes thionnh ., , )()1011S |)Iu , 
01 any other thiottlmo device, and p is Lhc qmmlily of lual II J 
Avould sene to maiaUim Lhc ono, n<1 { (oinpcialuic, il it wei. 
supplied from outside dining the pioccss Eiom the above lesul 
it follows that Callendni's foinmla piovirtcs foi Ihe nueision of th, 
cooling effect -which is known to occur m leal gases When (// -|- 1 ) 
is greatei than b the expiession loi- p ,s posilivc, (he s is thei 
cooled by thiottlmg. This is the usual case But when (n + 1) c j 
Jess than b, p , s negative, the gas is then waimcd by throlllmo u 



vin I APPLICATIONS TO PARTICULAR FLUIDS 



323 



hydiogen is at ordinary tempeiatures, and as any gas will be if the 
initial tempciatmc is sufficient^ high. By laismg the initial tem- 

C 

peiature the quantity (11 -f 1) c is leducecl, since c = Inveision 

of the Joule-Thomson effect occurs when (n + 1) c = b, or when 
nc = c + b But, as we saw above, c + b is the slope of any 

fd(PV}\ ' 
isothcimal line on the diagram of PV and P, namely I -775 I 

\ (LI j 'i< 

Ilcncc if Ihe isolheimal slopes up \\ith a giadient steeper than nc 

Lhe Joule-Thomson died uill be a healing, if il slopes up less 

steeply than Lhis, 01 runs level, 01 

slopes down, Ihe ellecl will be a 

cooling It will be apparent fiom 

llie'sc consideialions lhat mcasinc- 

ments of the cooling dfeel linnish an 

impoitanl means of sell ling the 

\ allies of the constants in ('allendaTs 

equation, apai t lioni dnec I dc leinu- 

nation of Ihe isolliuinal IIIRS Cal- 

lend.u in fael .issnmes Ihal Iheco- 

\olumc b is equal lo Ihe volume 

which \\e>uld be occupied il Ihe gas 

weie all condensed to a liquid, and 

(hen calculates the values ol n and 

c I mm observations ol Ihe cooling 

died ' 

An ilhishalion ma\. help lo make 
some ol Ihe above pomls clear In 
lig !)5, which is a diagiam ol Ihe 
Amagal t\ pc, \\ilh PT and P lor co- 
oieim.ilcs isolhemials au skdche'd 
(not lo scale) for a <>as obeying 
Calkndar \ ecmalion Thev, aie, as we 
saw, stiaighl lines within Ihe langc to which the equation is 
applied AS is an isofherm.il drawn foi a tempciatme such that 
Lhc.va.pom becomes satuialcd at a moderate pressure, which is 
issumcd to be uilhm. the langc of pressure for which Lhe equation 
Iroldsgood Aceoidingly Ihelrnc J^issbiarght, up to Lhe saLura Lion 
poinL 6'. The ciuvcd line through S is a poition of the' boundary 
nrive, below which lies the "wet" region, where the beginning ol 




w 
w 



Pressure 

Aiim^ni isollic i n 

\r Id ('llllc'lldlirH 
U'l IHlH 



* Phil May , Jan 1903, p 87. 



324 THERMODYNAMICS ft-ir 

condensation would be repiesented bj' a \ r eitical straighl lint 1 , #ir, 
P as well as T being then constant AS slopes dowmv.uds, and the 
effect of throtthng,at that temper atuic, is lo cool I lie gas I'fi' is an- 
other isothermal, drawn for a lo\\cr tcmpoiuLiue, lo which I he same 
remarks apply. The effect of throttling rs si ill lo cool I he gas al I he 
higher temperature for which the hon/onlal isolhennal />/> is di.nvn 
( c = &) and at any temperature up to th.nl of ('(', which is I he 
isothermal corresponding lo the imeision ol UK Joulc-Thoinsoii 
effect, namely that for which (n + 1) c = b, tlie npwaid gi.idie'iil 
of CC being equal to nc At any highci lenipeiahiie, such .is thai 
for which DD is drawn, the upwaid giadicnl is slerpei and I he 
effect of throttling is to heal the gas 

207. The Specrfic Heats m Callendar's Equation The t \- 
piession given in Eq (26) foi K !t , in a gas that conloinis lo 
Callendar's equation, enables the specific heal <il const. ml pitssiiu 
to be calculated for any temperature and am piessuu willnn I lu 
range to which the equation applies, when llu \aluc ol l\ lt ' (,is- 
snmed constant) foi the qiven gas is kno\\n as \vc'll ,is UK < onslaiils 
of the chaiactenstic equation In oielei lo ohl.nn ,i conespoiKlinn 
expression for the t.peei fie heal atconsl.inl \ohini( il is most con- 
venient to write the characteristic equation in UK lonn 

PU - RT, 

where U stands for V b + c U is a Junction ol / ,md 7' only 
Differentiating with respect to T, keeping V eonslanl, 



rr ( dP \ T, (<IU\ 

U -,- + P( R 

\dTJr \tlTJi 



I-'M 



AT- 1, ' -\dT), \,ir 

But (g) _*_ e . , K , ,<';") .'" "("I. 1 ''' 
\al/ r rfjf / \dJ-l, (IT- /'- 

Substrtutrug these values, andrcmembeiing lhal ' = , ,, we obl.un 
from Eq (28), y M 

U'Jr'7'l 1 - 1 - (/) ( 2 ). 

Then from Eq (29), 

Pnc f ^ %nc\ __ Rm / 

(I 



vnr] APPLICATIONS TO PARTICULAR FLUIDS 325 

Now by Eq (23) oPChap VII, m any fluid. 



dV T 
Hence in a gas 01 vapour Lo which Calendar's equation applies, 

< 30 > 

In Integra ting we have to icmembci llutal conslanl tempeiaLure 
dU = dV. Accoidingly, 

Rue / nc\ 

A = I // 1 -- ) -|- constant. 

Willing tins in the i'onn 

*.- -'(- I-) HA".' (81). 

we sec that I he conslanl ol inlegiation K u ' is the limiting \alne of 
K,, when 7 J - 0, \\lnch (like K ' Ail 200) is iakcn as having the 
same \alue il all U mpualincs lo ulnch Ihc equation is applied 
NL\I, to (ind an evpussion Joi K K, ]i\ Eq (2S) ol Chap. 



He nee in a gas lo which CallcndaTs equalioii applies we obtain the 
lelalion , ,., 

I (52), 



by subslihiling I he \ahn.s aheidv found in l^<]s (2S a) and (22) 
Poi Ihcsc l\\o dil(< Kiilial eoc Ilieii nls 

In the lunil \\hen I' =- I he \olinnc 1 beeonies indcliiiilely gicat, 

Ihc kun . \ anishes, and 

K; - K V ' ~ jt (3.$), 

as we should c\pccl Itotn the fact I hat I his gas is I hen lobe regauled 
as peileel (eonipaie Ait ISO) II should be nolul (hat Ihc assump- 
tion lhal A'/ is conslanl letjuues I ha. I K ' should also be conslant 

208 The Entropy, Energy, and Total Heat, m Callendar's 
Equation To find an expiession lor the entropy we shall apply 
Eq (20) of Chap VII which is line of any fluid, 



320 THERMODYNAMICS [en 

Here, and in what iollows, the icicloi A, which is 1 /J (Ail 1 1), 
is mtioduced in older that heat quantities (including R) may be 
uumeiically stated in thermal units. 

By Eqs. (26) and (22), Ait 206, we have, using thcimal units, 

An (n + 1) cP , , L (dV\ _ R Anc 
K = T + * \dTj P ~ P + "T~ 

in any gas that obeys Calendar's equation Hence in any such gas 



AY An (n + 1) cP , R dp _ Anc 

d6 = -~ dl -I --- ^2 -- IL ~ p T~ ' 



(31) 



Integrating, this gives 

'LncP 
</> = K v ' log, T - R log, P - - + 7? (35). 



or </> = 7v ; log, T - R loo c P - t H - 2? (35 ), 

wheie 7J is the constant of inlegiation. 

To find an expiession foi the inteinal eneigy E we may mosl 
conveniently use the geneial equation (S) of Chap VII, 

dE = Ttty - IPcJV 
In a gas that satisfies Callendai's equation 



By substituting this and the value of <7(/> in Eq (31) -\\o h.n e 
dE = (K ; - R) dT -|- An ( nc dT - cdP\ 

\ ~*~ J 

= K v 'dT -And(cP) (3(,), 

fiom which E - A',/ T - AinP + B' (37), 

wheie R' is the constant of inlcgialion NoU I ha I Lhr inUiual 
eneigy falls shoit of the value il would have in a peifccl gas by 
the amount AncP. 

To find the total heat we have, by definition / -- E -\- APF 
Hence, fiom Eqs (37) and (20 a), m a Callendai gas 

/ - (K v r + R) T - A (n -1- 1) cP + AbP -h B' 

= K P 'T - A [(n + l)c-b]P + B' (38) 

Fuithei, since A [(n -f 1) c b] is, by Eq (27), equal to the Joule- 



vnr] APPLICATIONS TO PARTICULAR FLUIDS 327 

Thomson cooling effccL p cxpicsscd in theimal units per unit chop 
oi picssure, we may wnlc this expression 1'or the total heat in the 

foim I = K,'T-pP+B' (38 a) 

On diffeientiatmg Eqs. (37) and (38) with respect to P, keeping 
T constant, we have 

Anc and ~ A 



These icsults agicc wilh the cxpicssions ahcady gi\cn iov the 

cooling effect The whole cooling effect is - (-. = } . by Ail 182 

\dP' T J 

it is made np of 



Fin thci, since f J is constant foi any one tcmpeiatuie, the slope 

of any constant-tcmperatuic line is constant., on a chait of / and P 
foi steam (compaic AiL 102) The hues slope downwaids, with 
ineicasmg P, and llic slope is less at high tcmpcialnics, since < is 
then less 

To complete Ihe list, expulsions mav be added foi the Junction 
(\\lucli is G) and Hie hmction i/r, in a Callendai gas These aic 
found at once horn Ihe above lesults 



= A'/ T (1 - l<,n ( 7') -i- RT IOO L p-j( c - I,) p __ HT -I /?' 
ifi = E- T(f> - K,,' T - K J T lon i T \ UT 1< )i> t /' - JIT -i //' ( 10) 

All the foieoomo <h duel ions liom C'.illc nd.n's cf|iialion hold good 
foi any gas 01 \apoin to which the C'(|ii<itiou applies, \vhalc\ei b( 
Ihe \alucs ot the constants, piovidecl the specific heal at /eio i)ies- 
sme may be taken as independent, ot the tempei.il me within Ihe 
langc ol apphe<ilion 

209 Application to Steam In applying Ins equation to steam, 
Callcndai assigns to the constant n a value such that 

nil = K v ' 

This iclalion, which is not true for all gases, gives a value ol n Joi 
steam thai is consistent with the ob.se ived el'lccts of tluotlhng It 
has the practical advantage of allowing expressions for the bc- 
haviom of the gas dining adiabahc changes to take a veiy simple 
form 



328 THERMODYNAMICS [en. 

When K v ' = nR it follows fiom Eq (33) that Ay = (n + 1) R, 
and in that case the expiession for cj>, Eq. (35 a), becomes 



= ( n + 1) R log c T - R loo t P - 
P 



(41). 



Now in achabatic expansion lemains constant, and LhaL can 

p 
happen in this expiession only if - hl is constant Hence in the 

adiabatic expansion of a Callcndai gas in which the i elation 
nR = K v ' holds, the piessuie and tempciatuie aie conncclcd b} 
the equation p 

yw. = constanl (42) 

p (Y _ M 
Fuither, it follows that in all such cases - - is constant 

dming adiabatic expansion, because by the charactcustic cqualion 
we have P (V - b) _ P 

T ~ T >1 " l ' 

and under the condition stated both leims on the ii<.'ht-h<ind side 
aie constant. 
Again, undei the same condition that nR = K v ', 

T n *P(V - b) 

-p- - T = conslanl, 

I 
whence T (V - b}" = conslanl ( l.J), 

i u i i P(V-b) 

and, multiplj ing b> -- - , 

n j-1 

P(V - b) " = constant (IJ) 

All these Jesuits for adiabatic expansion aie line of steam, willnn 
the limited lange thiouoh which Callendai s cqualion is ai>])hcal)lc 
They hold good so long as the substance icmains in the homo- 
geneous state of a gas, uhdhei supeiheatcd, salnrated, 01 super- 
cooled (Ait 79), and they cease to apply when pai t oi it liquefies 

In the calculation of his steam tables Callondar lakes foi Ihe 
numeiical value of the co-volume b the volume ol unit mass of 
water at C , namely 01G02 cubic feet per Ib. Foi R he takes 
11012 in mean caloues, corresponding to 1-982 per niol, and 
equivalent to 151 17 foot-pounds pei Ib Foi n he takes \ This 



vm] APPLICATIONS TO PARTICULAR FLUIDS 329 

fig me is based mainly on thiottJmg experiments by Ginidley **, 
Peakc[, and Callcndcir himself | 

lie lakes fov C a value such that c is 4213 cubic loot al 100 C. 
This makes 

157 52 x 10 

C=Q 1.213(373 1) '', or 15752x10, and c = - |in 

Ilcncc when V is the \ olumc of 1 Ib. in cubic feet, 

P is the picssuic in pounds pei squaic foot, and 
T is the absolute tcmpciatme in ccntigiade dcgices, 
the Callendar equation j>y 

r = - c -i- b 

becomes, for diy slcam in any state, 

1 T I 1 7 7' 1 T7 *V> v 1 

F = - f -I-001C02 (L5). 

p r * 

As a numciical example, let it be lequned to find the \olumc of 
1 Ib of steam at a ])iessiue of 100 pounds pci squaie inch and a 
L-mpeiatmcof 2 10 C. lieu P = 100 x 111 and T = 513 1, making 

}' = ] ;J733 - () I ISO -|- Oll0 - 1 2J37 cub It. 

This will be found to agice \villi the value m Table C ( Yppcndix III) 
\\hcic lhe\ oh i me is labuLiled toi vanous picssuus and for tempcia- 
tmcs langmg fioni 100 C down lo llic Icmpeialmc of saturation 
and belou it The \ oliinu's below llic kinpciahueol satmation icfei 
to \\ilei-\ajK)iii in a supc icooltd (melastablc) sttile, such as that 
which is sol up by adiabatic e\jiansion in the absence ol nuclei on 
which condensation niav occm. In this example the slcam is 
shghllv siipc t he alcd, llic salmalion lemjieialuie loi a piessuie 
of KM) pounds bcmn pisl uiidei 230 C 

Callcndai also labiiLtli s sepaialt l\ I he "co-Jg>icy<ition \oliime" 
c loi \aiious Umpi'ialmes Some ol I he values aie yivcn In. low 

volume c /( Dn/ Steam in tint/ ttffilc, in 
cubic led [)c\ Ib 



Pomj) 


i 


1Vni|> 


( 


Tom IP 


i 


Toin[) 


f 





I 102 


70 


5570 


1 10 


1000 


210 


1780 


10 


1 () r >7 


SO 


o 5o<>i 


150 


02771 


220 


o i <)<:* 


20 


00117 


00 


K.I! 


1()0 


25<>2 


230 


1555 


30 


H4JO 


100 


1213 


170 


2375 


210 


1 150 


10 


7557 


1 10 


o 3sr>7 


180 


220J 


250 


OlSdG 


50 


OSO 1 


1JO 


WO 


190 


2010 


2(50 


12S2 


00 


0147 


130 


3255 


200 


1009 


270 


1205 



* Phil Tintit A, vnl 10 1, p 1,1000 | Pine Tioif Sot A, vol 7fi, p 185, l ( )0fi 
| ScoBunkwotUi, Phil Tunis A, vol 215,]) 383,15)10 



330 



THERMODYNAMICS 



[en 



la furthci illustiation of Eq. (J5) vanous isothcinial lines Jtoi 
steam aie diawn to scale m fig. 96, showing PV in lelahon Lo P as 



500C 



800 



700 



__400"C 



600 



: 30gc- 



500 



400 



300 




200 



100 



Pressure, Pounds per Sy Inch 

100 200 300 ~400 500 

Fig 9(3 Isotheinuils for steam, fiom Callciul.n'H eyiuiLioii 

calculated from that equation Here the pressuies aie expiosscd 
in pounds per sq inch, consequently the nuniencdl values oi PV 



m] APPLICATIONS TO PARTICULAR FLUIDS 331 

jven 111 the figure must be multiplied by I'll li it is dcsned to ha\ i i 
hem m foot-pounds The dolled continuations of the isotheimal 
nes ioi 200 and 100 below the satuiation cmve lepiesent values 
31 supei cooled 01, as it is sometimes called, supcisatmated vapour 
'he full lines diawn at constant piessurc lepresent the first stages 
i the condensation of a wet mixtuie It will be obseucd that at 
he highest temperatuie at which Callcndai applies his foimula 
a steam, namely 500 C , the isolheimal still slopes down with 
ici easing P Thioughout the whole woilcmg range the thiottlmg 
f steam produces a cooling effect. 

Since the value assigned to R it> 11012 caloiy, and that of n 
, \- 3 the i elation K v ' = nR icqunes that K e ' bliall be 036707 
nd A"/, which is K v ' + If, shall be 17719 

We have next to show how the tabulated values of the total heat 
ic calculated The foimula for /, Eq (38), becomes, Ioi steam in 
ny homogeneous condition, whctliei supci healed, saluialcd, 01 
upeicooled, j ^ R ^ T _ { ( i^ _ b} p ,. Jr 

ivuio., in caloncs pei II) , 

/ = () 1771!) T _ C'ArJi) 7 ' .| u> (|0), 

hue c and b ate expiesscd m cubic feet pci Ib ind P in j)oiinds 
ci sq loot 

To obtain a numeiical value Ioi /?', which comes in as a conslaiit 
f mki>iation, we musl fix some /uo slalc fioiu \\hiih Ihc lolal 
( at ol the substance is lo be icckoiiLcI, 01, whal comes lo Hie same 
img, we must assign a uuuKiieal \alnc lo llu lolal heat m SOUK 
nown slate In llic calculation ol lus l.ibks Callt ndai assume s that 
le tolal heal of \\atei is /eio at C and is 100 al 1()0 U C 1 , nuclei 
ituialion pressuie m each case ' 

* r l'lm insiniipluHi nnt onl\ li\is llic /(uo fidin \\lmli lliu Ldliil liml IN tu IK 
c'koncd, bill alyn i;i\os Lo tin 1 thci nial mill n \aluo \ciy slililly ^noaloi lluui Ilio 
iMii onloiy ,;^ flc'liiad in Ail 13 The tluiin.il anil ol ( '.illi luku'w lahlis and 
'inuiliiH is one liuiirlicdlli of I hi 1 Llnin^c in lnl.il heal \\\\u li w.ili-i iiii(lci()c-< \\lii n 

n licak'd 1 1 (Jin () Q (n I OK 1 midoi Uici (rai yniu) ii< ssmo ol nalu ration, \\lic KMH I ho 
duiaiv moan ca-loiy is OUP ImiulKHllli of Ilio change' in liil.il Jic^at \\liin \\.ilci i^ 
iatod lluoui,'li I hi 1 aiinio inLeival of (.oin|)0i,iimo nndci a loiiHtunt [inwsinc ol 0110 
.mosphcro Calk nclai's unit" 11 tin. laigei of the two by about one part in fom 
lousiuul Thin difleinnce i ol no pi actual nnpoitancc il IH so Hinall as to bo \\ill 
rtluu tho limits oi CHOI ot cvpeiirnont The fifruios Ioi Ilio mc'( hanical t'limvak-nl 

In'al gi\c-n jn Ar t I i i plate, slu< tly, to Ilio laigoi unit, whu b is tho one iiHc'd in 
to tablos Tallondai takes his caloiy to be equivalent lo -4 ISdK / 1U 7 CI^H, tho 
11 1 csponding value ot the constant piosmne moan caloijr would bo J 18H8 x 10 7 oi{j;s 

Tho relation between the two iimtM will Ijo intido i loai if we \vnlo out an onoig} 



332 THERMODYNAMICS [en 

In passing fiom the stcite of water at 100 C In thai ol di\ 
saturated steam at 100, undei a constant pressuic oJ' one atmo- 
sphere, the fluid takes in 539 30 calones, that being the latent heat 
as determined by experiment. Hence / foi steam A\hen T is 373 1 
and P is 14 6S9 x 144 is 639 30 The value of c at thai Icmpcialme 
is 4213, and b is 01602. Substituting these figures in the ex- 
pression for /, Eq. (46), we have 



639 3 - 1TT19 X 373 1 - . 14 + fl , 

fiom which B' = 463 995. Foi most purposes B' is taken as 464 

The foimula foi the total heat of si earn, m any condition within 
the Callendar lange, accoidmgly becomes 



- 719 T - - -I- 4.C-1 H) 



Values of / calculated in this way aie given in the tables As an 
example, take the same case as befoie, namely steam al a picssme 
of 400 pounds, per sq inch and a tempeiatuie of 240 C With these 
data the numbeis aie 

/ = 241 84 - 25 20 + 464 = OS3 55 
The tabulated value (Table D) is 6S3 54 

account for the piocess of wanning water Let i 7 ,, be the internal e IK ILT\ of unit mass 
of water at C and the conespondmg saturation picssme P () , whidi is H02 / UJ 
pounds per sq ft , and let / be the total heat in that slate /=/,' I J /' \ Lei 
E IM be the internal eneigy of watei at 100 f and the coiicspondmg s.itninhon 
piessuie P 100 , which is one atmosphoie 01 14 GS9 , 111 pounds pei sq It , and h I 
7 ino be its total heat ui that state I im =E 1M +Al\ (M ] Inn I n is (lie \olnnii al, 
0C , namely 016 cub ft , and V 1IU is the \olume at 100 Imamo tlio wnlci, 
initially at and P to be uncl^i a piston Incicaso the load on tlu pislon lill tin 
piessure is P )00 Since watei is almost inconipicsMblc this cloc-H not Hensibly cluing 
the volume, 01 the tempeiatuie, 01 the intunal cnng}, wludi may be (akin an 
still equal to E (l Then heat the w, ate i nuclei constant picssiuo to 100" (Ins it-qmiLS 
the addition of 100 constant pressuic caloiies In being heated the watei uvpaiidn 
from V to F 100 and thciefoie does woik on the jjiston equal to AP l{m ( V im - F ) 
Hence the net gain of inteinal eneig^ in the whole opeial ion, (<\i>irss( fl in connlant- 
piessure calones, is r _ 1ftl }) nr 
from which 

or 7 ino - / = 1 00 + 1 ( P 100 - P ) F,, - 1 00 023 

Thus the same change of total heat winch is measuicd bv 100 ol C'allc-ndai's units is 
measured by 100 023 constant-pleasure units 



in] APPLICATIONS TO PARTICULAR FLUIDS 333 

It follows from Eq. (37) that in steam 

10 cP 

= 36709T - + 164 (47). 

'he above expicssions are in tciins of P and T. We may also 
vpiess both E and I foi steam in teims of P and V, eliminating T 
mcc K ' = nit, Eq. (37) may be written 

E = nRT - AncP + Ii' 

nt by the c'hriractenslic equation (20 a), wJicn R is expiessed m 
icimal units, RT AcP = AP (V b) Hence, in steam, 

E = AnP(V-b) + 13' (18), 

Inch gives, in thennal units, 



Uciin /, being equal to E -+ APV , becomes 

/ -= J(n + 1)/'F- AnbP \- B' (J9), 

ving, m thcinial units, 

i \pv - o w 



-(-JO I- (19). 

100 V ; 



This leluLion nitiv be wntlcn in the foim 



00128 (W) 

:ncc if we use /> to denote the piessiue in pounds pci squaie incli, 
e volume, in cubic feet, of I Ib ol steam in any diy state, supei- 
atcd, satuiated, 01 snpeicooled, is given by the formula 

v _ 2_2i.36 (1 - 404) + 

This affords a convenient means of calculating the volume when 
c total heat is known. Take dgam the same example as befoic 
am at 100 pounds per squaie inch and 240 C The tabulated 
hie of / is 683 54. Substituting it m the formula we find V to be 
J314 + 0123 = ] 2137 cubic ieet, in agreement with the figure 
t from Eq (45) and with the tabula led value of V 



334 THERMODYNAMICS [en. 

By differentiating Eq (49) with icspect to P, keeping V constant, 
we obtain, in steam, 

1T\ 

= A(n+l)V-Aub (51), 

which is constant foi any chosen value of F It follows that lines 
of constant volume on a steam chait with P and / foi cooidmalcs 
aie stiaight in the legion of supcihcal ', as in fig. 33 (Ait 102). 

We shall next obtain a Moikmg loimula foi the cnhopv ol steam 
in any diy state, b} using Eq (35) and finding (he value ol the 
constant B The constant is found bv woikmg ou(, horn indepen- 
dent data, the entiopy oi sal mated steam at 100 C 

Following the usual convention the enliop\ ol icnfci at C is 
taken as zcio It follows fiom uhat is knoun about the specific 
heat of watei, as will beshoivn in thcne\l ailicle, that the enliopy 
of watei at 100 C and a picssuic ol one atmosphere, is 0. Til, So 
In passing at that constant piessmc liom the slate ol walci <il 100 
to the state of satin a ted steam, the substance takes in 5.19 .5 units 
of heat at the absolute tcmpeiatuic 373 1 its cnliopv the le loir 

539 3 

mcieascs to 31 ISO + -- oi 1 75732 At that lempciatuie c is 
373 1 

4213, and P is 1 1 GS9 x 1 H Hence, by Kq (3 j), 

1 75732 = 17719 log, 373 1-0 11012 log, (1 1 689 x 1 1 I ) 

_ 10 [2]3 - " '' ()SO "I" 

3~ 373 1 y 1 100 ~ ' 

Flora which B = _ 021901 

Substituting this, and inlie)diicing the f.uloi 2 3025S5 to comeil 
common to Napierian logaiithms, the Jonuula Joi the cnliopyol 
dry steam m any state becomes 

= 1 09876 Iog 10 T - 2535(5 log 1(J P - 0023S1 ' - 21901! 

(52) 

Values of d> aic given in Ihc tables (Table K) fei the same range of 
piessme and tempeiaturc as was used m tabulating T and J. 

210 Total Heat and Entropy of Water It is known from 
the icsearches of Regnault and others Ihat Ihe specific heat of water 
is not constant, but increases with using temperature. Callcndai 

* In the wet region tho constant- volume hnoa leinam voiy nciuly shaight, for 
tho above relation still holds with legaid to that pait of tho steam which is uncon- 
densecl, and its volume constitutes neaily the whole volume of tho wot mrctme 



in] APPLICATIONS TO PARTICULAR FLUIDS 335 

uggests* that this inciease may be due to the piesence of water- 
apour dissoh r ed in the watci. He supposes that when watei and 
s vapoui aie in eqiulibiiuni at any tempeiatinc a volume of the 
a pour equal to the volume of the watei is contained within the 
r atcr Consequently when water is heated its total heat mcicascs 
lore lapidly than it Avoulcl do if the specific heat weie constant, 
>i the heal that is icqmrcd lo I'oim the dissolved vapour becomes 

laigei piopoition of Lhc whole heat This thcoiy gues icsults 
r hich aic consistent with Ilie experimental d.ita, and Callcndar 
dopls il in calculating, loi his lublcs, Ihc total heat and the 
nhopy of walcr It has the aehantagc of allowing each of these 
wo quantities to be expicssed in a simple manuei 

Lcl J\ be the volume of 1 lb of sahnatcd steam al anv assigned 
'mpciatuie T, and let V m be the volume ot 1 lb of walei at 
ic same lempeiatuie and picssuie Then accoidmg lo Ihe 
u'oiv, 1 II) of *' walei" in that stale is leal!}' 1 lb of a solution, 
)nlaimni> chssohcd vapom , Ihe walci conceals willim it a 
olumc of sahualcd steam equal lo /' ', If the icmamdei \\eie 
Ko tinned into vapoui, undei coiiilanl piessiuc, we should June 

tolal volume of vapoui equal to V m + (J\ - J r w ) 01 V\. and 
ic heat taken in dining Ihe piocess would be (he latent heat L 

LV , 
[(nee "' icpicsents the heat that is icquued to pioducc Ihe 

' S ~ * II' 

ipoui iiiihallv pies(iil m the walci bcloie Ihe 1 loiuiriiion ot any 
pai.ili st(am begins This heal had lo be supplied \\lulc the- 
alei was berny wanned up lo Ihc tempeialuu ol satiualion, il 
insliluU^ a pail of (he lolal heal ol walei /, 

Tin olhei (and elue-l) pail of (he. lolal heal ol w t itei i 1 - supposcel 
> HICK asc at a iiiiilonn lale is Ihe le mpeial me i ises it may llicie- 
ne be leptc'smlcd by h. (T T (] ), where it is a conslanl anel 

7' is Ihc excess of lempcialmc ,ibo\c (' , \\luch is lakui as 
10 stailing-|)oint in icckoniii" Ihe te>tal heat Hence, adding Ihc 
vo ])aits, Ihe tolal heat of watei undci satuialion piessme al 

ly LempeiaLine T is 

r 7-r TV 

~~1"- __ "'" 



T \ \ 

- 1 OJ "I" 



TT ir ir T 

y ' ' ' 



cie L n , F H)O and F s i cfei to the state at C. At tha t tempeiatuic 
ic latent heat is 59 I- 27, the volume of water is 016, and that of 

Luraled steam is 3726 cub ft Hence v ~~_- -= 0029 calory 

SD io 

* Phil Tians A, vol 190, p 147,1902 



336 THERMODYNAMICS [en. 

This is the constant which has to be subtracted to make I u = 
when the temperatme is C. 

To calculate K, we have /, = 100 when T - T = 100, by Ail 
209. L is then 539 3, V w is 01071 and F u is 26 789 Hence 

_M9 Sx 01071 _ 



from which K = 9966(5. The foimula Foi /, tlius becomes 

I w = 99666 (T - 273 1) + J^jf ~ <> <><>-} (53 a) 

" > 'u> 

Values of /, calculated by this foimula aic given in I he tables. 

V 

Thioughout the woikmg lange the ratio _'" is vciy neailv 

' S ' !(, 

the same as V W \V ^ and ils numci ical value is appio\imalclv 
equal to 00004p, wheic p is the sahualion piLssiiu in pounds 
per sq. inch 

To find the entiopy of watei undei salutation picssmt <il <mv 
tempeiatiue T, we may think of the walci as bung biounhl lo ils 
actual condition by two slciye.s Imayine it to be hc.ilrd lf> lluit 
tempeiatuie in an '"ideal" mannei, naniclv \\illiouL I he loim ilion 
of anj^ dibsolved steam, and then the dissohed sic am lo IK lonned 
at that temperatuic The entiopy depends only on UK ,11 lu.il con- 
dition (Ait 14) Taking, as bcfoie, the cntiop\ ot \\alu at C' 
to be zeio, we theicfoie have 



= 99666 log, ~ + _, /5- Jl V , ~ 00001 (5 I ), 

* (' i, ~ ' ial 

which is the foimula used by Callendai foi the enliopy ol watci 
undei satiuation picssme' 1 

It follows that the value ol G, 01 TV/> /, loi walei undei sahna- 
tion pi assure is 

G w = K T log f ^ - OOOOlT - K (T - T () ) -\ 003 (55), 
* o 

the term v l " cancelling out 

" S " 10 

211. Relation of Pressure to Temperature in Saturated 
Steam. A formula connecting the piessurc with the lempeiature 
of steam in the satiuated state is rnosL easily obtained by making 
* Sluim Talks, p 7. 



[i I APPLICATIONS TO PARTICULAR FLUIDS 

j of the fact tJial G, 01 T(j> - I, has the same Milne toi the 
urated \apom, at any tcmpcialme, as loi the liquid at the same 
npciatwc and picssiue (Ails 90 and 185) Ey Eq (39) of Art 
8 the \alne ol G foi steam in any state is 

Ay T loo. T ~ Ay T - HT Ioy t P + A (c - b} P + BT - B' 
nee for diy steam at saturation pi cosine 7 J S 

- A'/ T loo. T - A'/ T - RT loo. 1\ + A (c - b) 1\ + BT - B' 

(50) 
ice G b - G to \\e oel, by equaling (55) with (5(1), 

t i . li\ p 

~ - B -I- /c - Ay -I- K loy e T + 00001 



oi> - 



This c \piesMon allows the saLuiaLion piessuie P., to be iound lor 
/ lcni])t i aline On t>mn<.> the \anons eonstanls the values 
eacly slaletl, il becomes 

101 J l(n i P - V ' " = 5 S!JO!)I - " ' - - 51') 17 loo. T 



lendai ' pills llns m a loim moie smLibk loi e.ilenl.ilion, b\ 
(sliditmy 2 o()'2,")S'j lo^,,, loi lo^ L , and 1 I ly;, loi l\, />, biin^ I lie 
malion pussiiiL m pounds pel squaie inch 

105? (< - h] /> 2)()'J .5') 

i/'s~- \,, - J107II')- - I 717.51 li)i) |(| 2' 

(57 h) 

1 s.ilmalion pussine eoiiesponduin lo any oi\tn Umpti time 
ioiind b\ woikniLj out the iinhl-hand side ol llie eqiulion 
1 then .ul|iislinn MK \aluc t)t loy p^ unlil llu l\\o sides bceome 
lal. 

'he picssiiu s ol satmaled sltam, ihns cl educed J'lom (.'.ilIendaTs 
lac'lcii-ilic ecpuilion, ayiee \ r eiy closely, Ihioii^houl Ihe lanyc 
ivhich I lie equation applies, with those mcasmed by Rennault, 
I Ihe anicement beh\cen the ealeulaLed and measnied iii>mes is 
lenee ol Ihe soundness ol CallendaiAs melhod Finlhei con- 
lalion is oblamed when the \olumes, as calculated by Ins 
alion, aie conip.ucd with experimentally niuasuicd volumes 
h of sal mated and ol snpcrhcutcd steam 

* MLUIH Tahiti,, p 27 
E T 22 



338 THERMODYNAMICS . [en 

212 Formulas for the Latent Heat of Steam, and for the 
Volume of a Wet Mixture Fiom the equations 

L = I, - I a and / = Kt + ~ "p- - 003, 

' s ' w 

where / is the tempeiature on the centigiatlc scale, we lui\ e 



or L = (I, - ><T + 003) (l - p"J (58), 

which Callendar p wntes 

V\ 1 

chopping the 003 as negligible in this calculation 

For the \olume of a wet mixtuic, V v (Ail 71), hi gives the 
foimula ' V I Kt 

F / -// (r>0) 

To obtain this ^ e ha\ c 

M'w 

*i ' //' 

on again dioppmg the 003, also 

1, = L + I al = L + id -l- ' "' 

' s ' ii' 

I f . Kt qL (V ^ T lr } H- />/ 
Hence _ f = J ~<~\,'~ ~\^ ~\ I / r 



r 



213 Collected Formulas for Steam For c'on\ UIH-IK-C ol 
icfcicnce and use tlic foimnlas aic oolleolc-d IICMO h\ r nuansol \\Inch 
the quantities in the Steam Tables may be cah-ulaU d 

In these foimulas V is the volume ol 1 II) in cubic Jcct, /* is Ihc 
picssme in pounds pci squaic foot, and p in pound 1 , pci squaic 
inch Centigiade degrees aic used in the reckoning ol Icnipcialnrc 
and quantities of heat T is the absolute lenipeiature and / the 
tcmperatuie fiom C. 

The following expicssions foi V, /, E and r/> apply lo chy steam 

* ftlram Tablet,, p 10 



air] APPLICATIONS TO PARTICULAR FLUIDS 339 

n any state, that is to say, supciheated, saturated, 01 supeicooled, 
bul not to a mixtuic of steam and Avatcr 

The volume 15 , ,,., ,-- r> UG 

Vss l*Ll.J _lo/o-xlO 

^, 2 ,';- ^ ^ 

The lot.il heat ,, , 



40 1 . (10 a). 

.vheic c is Llie ' co-agyicyaliou volume" in cubic feet, namely 
15753 XO" ,373 ' 



T ' V T 

EquaLion conned my the volume \viLh the picssuie and total heat- 

r = ** l V--) + 00123 (06). 

y> 

The internal encigy 

\()P(l' - 010) 

,,, , '"- .. n M01 < 1So > 

I lie cnlio|)v 

cP 

/, 1 0<>S?<>l<>n l() r-0 J,5.J50l()o 1(J / J -0 002381 -0 21964- (52) 

The following c({ii.ilion, which applies only when the sleani is 
il salnialion [ju'ssine, deltimincs the icLihon ol picssiuc lo 
lempualme in s.ilui.ited slcam 



>( )03 .V) 

- '21 071 l ( > - " - I 71731 lo"!,,'/ 1 ( r vr/;). 

\Vlitn tlie satmalioii piesMiie lot an\ i>i\en tcmpcialuic T has 
been delrimincd l)\ nn.uis ol ^lns einialion. Hie \olunic, lolal lieat, 
_ > ncii>\ ind enliopv of salm.iU'd slcain at that lenipciatuu (l'\, 
I , J<^ and c/> 4 ) aic, lound b\ Ihe abn\c loinuil.is 

The lalcnt heal 

L =(!,- f )00(>0/ -I- 003) (l - '" ) (58), 

\ ' s ' 

\vlu b ie V m is the Aolunie of 1 Ib of walci at saluialion ])iessuie. 
\Vithin the ranye usual in piactice, Ihe lalio V w \V a is veiy ncaily 
.qual to 0000 ly; s , wheie p, is the salutation piessure in pounds 
pei square inch, and the working ibimula is 

L'-=(I o -() 990CG/) (1-0 0000 l-p a ) 



340 THERMODYNAMICS |c 

The total heat of watei at satuiation piessme 
/, = 99G66/ + j^^jr - 003 

' X t, ' in 

01, veiy ncaily, withm the working langc, 

I IL -= 99666* + 0000 lp,L 
The entropy of \\atei at saturation picssruc 



The function G, uhich is Trj> - I 
Foi chy steam in any state, 

G = K ] ;Tloo t T-(K ] ;-B)r-l!T\o^P + AP(c-b)-Jr (39), 
which gives 
G = 1 09S77T Iog 10 T - 696832' - 25350?' loi> 1( , P 

+ *<'-'>">_ l()l 

I 100 

Foi satuiatecl steam, 01 \\aLcr at saLuialion pic-ssuic, 01 a niixlino 
of uatcr and steam in eqiuhbinmi, 

T 

G b = G, e = K T Ioy t - - : (T - T n ) - 000017' | 003 (55), 

* o 
which gn cs 

G, = G w = 2 29I9T lou 1(J ^ _ o 9)0(i(,/ - ( ^"'J' 

214 Tables of the Properties of Steam Tin Slr.im T.il>ks 
in the Appendix contain some lepicscnlalivo nunihc-is, l)iil iclci- 
ence should be made to Callendai's Tables I'oi a nioic cout])hlt scl 
Tables A and B relate to the special case ol MtLiualtd sli'ani A\'hc n 
steam is satuiated a single piopeily, such as oithc'i I IK- Icnipcialuic 
01 the piessine, fixes ils state In Table A I he j>iop(.ily which is 
assumed to be kno\\n is the tempeiatmc, and Ihe labk' yi\ cs coi ic>- 
spondmg values of thepiessiuc, volume, lotal hc.il, and cnliopy 
all foi the satuiated state It also gives the latent heal Snmlail} 
Table A' gives the volume, total heat and enliopy of ualei at 
saturation piessuie It also gi\ cs the funcliou G, which is Ihe same 
for water and for saturated steam. In Table 13 Ihe piopei ly which 
is assumed to be known is the piessine, and coiiesponding values 
aie given of theothei piopeities in the saturated state, namely the 
temperature, volume, total heat, entropy, latent heat, and the 
function G 



vmj APPLICATIONS TO PARTICULAR FLUIDS 341 

Tables C 5 D and E vela I. e to the geneial case ol steam in any dry 
5 Late, whether superheated or snpei cooled A knowledge of two 
propcitics is then icquncd to specify the state, the two that aie 
Delected as independent variables in the tables aie the tempciature 
ind the picssmc. Table C gives Lhe volume, Table D the total 
icat, and Table E the cntiopy, in i elation to these two variables 
[n each table a, zig-zag line indicates (he bonndaiy between Lhe 
aipcihcatcd state (abcAe) and the supcicooled state (belou). In 
'tossing this line the substance passes thiongh the state of satura- 
1011 

Fiom Table D it is easy to find Ihe heal of foimalion, nuclei 
onstant piessuie, ol' steam m any condition of supciheal The 
otal heat, at tlie gneu picssme and tcmpcialiue, is obtained fiom 
he table, and Ihe heat of IbunaLion is found bv subtiactnig liom 
hat Ihc total heat of watci, al Ihe same piessuie and at Ihe 
empeiatuie ol I he leed 

Again, liom Table- 1), \alues mav be infcncd of the specific heat 
K ;( ) ol steam <il U mpe lalmes and picssiucs \\itluu Ihe uingc of 
he table- K tl leu <in\ condition of the steam is equal to the amount 
}\ Mhich / mcicasLs pel degie'e ol use 1 111 Icmpc lal me, at constant 
)iessuic The change ol / pe'i ek'giee is louud bv noting, m the 
ppiopuatc piessuie column, Ibe 1 amomil b\ \\hich / change's foi 
u mlei\al ol 10'\ and ch\ idmg lluit b\ 10, this ^i\i.s the mean 
alue of K tl o\ e i the mUtxal, which is pi act icallv the same Hung 
s K lt at the middle 1 lempe laluie N'aluc-^ of llu specific heal al 
anous constant |)it ssiues .mdloi \ <u ions lempc laliues li.n c been 
educed m (his \va\ r liom I lie tabulaled x.ilui.'s ol Ihe total heal, 
nd aie gi\en scpai.ileh m Table I ( ' 'IMie /ig-/ag hue IKIS Ihe same 
leaning as in Ihe olhci tables, llu- ligmcs abo\e it i< l.ilc lo siipci- 
caled sleam They sho\\ r a dcci ease ol A^xvilh using Icmpcialiuc, 
ut at luglu'i lempe latuies (bcvond I he lange lo which Calleiulai's 
q nation applies) Iheie is a maiked incicasc, as \vas pomleel out 
i Chap VI 



APPENDIX 1 

EFFECTS OF SURFACE TENSION ON CONDENSATION 
AND EBULLITION 

215. Nature of Surface Tension In Ails 135-138 it was pointed 
out that when watei -\apouv is suddenly expanded iL assumes a 
mctastable state, becoming supci swindled owing lo wh.it was 
there called a static ictnidation m the I'm niation of diops Wilson's 
experiments weie cited to show that, in Hie absence ol' foieion 
nuclei, a vapom will become much supcisaluialed l>cToic chops 
will form, and it M r as mentioned thai I his is an cITecl ol sin lace 
tension in the liquid In tins note some account will be gi\cu of 
what is meant by sin face tension, and how il ulaids [he loiinahon 
of drops in a cooled vapoui , also how it letaids Ihc loiinalion of 
bubbles within a liquid when the liquid is boiled 

The cohesive foices which Ihe molecules of any liquid c\crl upon 
one anothei make the fice sin face of Ihe liquid Inline as if il 
weie a stretched clastic skin. Il is to I his lhal Ihc phenomena of 
capillanly are due the use of a liquid column in a lube \\hen Ihe 
liquid is one that wels il, and the dcpicssion of Ihc column \\hen 
the liquid does not wet the tube To this also aie due Ihe hums 
assumed by liquid films and by chops II is Ihc lension of Hie 
surface layei that makes a chop lake a sphcncal shape when Iheic 
aie no chstnibing foiccs the chops of mollcu melal in a shol- 
tower, for example, become sphcies as they fall ficcly, and solidify 
into sphcncal shot befoie they icach the bollom A chop of mei 1 - 
cuiy on a plate, 01 of clew on a leaf winch il does nol wcl, would 
be sphencal were it not foi the upwaid picssuic of the suppoil 
on which the chop icsls, the smallci Hie chop is Ihe neaici docs it 
come to being a spheie, foi the distmbing foice due lo the weight 
is relatively ummpoitant in a small chop As a resull of smface 
tension, the cneigy contained in a chop of liquid is giealei lhan 
the eneigy contained in an equal quantity when lhal loims pait of 
a big mass of the same liquid at the same tempcialuic, foi energy 
is stoied in the surface layer in much the same A\ay as it would 
be stoied by the sti etching of an elastic skin 



u'P.i] EFFECTS OF SURFACE TENSION 3i3 

We are concerned heie only with theimodynamic aspects of sur- 
dcc tension, and especially with its influence on the formation of 
hops in an expanding vapour. We shall see that, as a consequence 
if suiface tension, a small chop will evaporate into an atmospheie 
if snpci saturated \apoui, because the vapour piessure which is 
cquncd to picvcnL cvapoiation fiom the cui\ed suiface of a diop 
s gicatci than the vapoui prcssuie which is sufficient to pi event 
vapoiation fiom a flat suiface of the same liquid at the same 
cmpeiature, in otlici Avoids, that at any given tempeialure the 
aluuition piessuic foi a small diop is greater than the noimal 
atmation pressure. 

The film that is fonned when a soap-bubble is blown, 01 when a 
oapy liquid is smeaicd over a ung or hoop of wire, consists of two 
ui face laycis, back to back, with some of the liquid between. 
Vhcn the film is vciy thin, as, foi instance, when it looks black in 
cficctcd light just befoic it breaks, it may be said to consist of 
\vo suiface laycis onh , but it can be made a hundred 01 moie 
imcs thicket than that and ha\e just the same tension, foi the 
tate of tension exists in the sin face la>eis only The tension of 
uch a film, \\liethei thick 01 thin, is the tension of h\o suiface 
iveis, in othci wouls, if is t\\ice the suiface tension The tension 
11 a liquid him chlfeis fiom fhat of a slictchcd sheet of mdia-iubbei 
i ollu'i clastic 1 mcmbinnc 1 m these impoilant icspccts it does not 
lunge \\hen the him c'outiacls 01 is si i etched, and il has nccessanly 
lie same \aluc in all diicetions along llic suiface 

Let a liquid film he I mined on a U-shapcel fi-une (fig 97) by 
kctling a wiie .111 with I he 1 
iqmd, placing it o\ c\ C\ and 

hen diawing il away in fhc , ^ 

luccLion of I lie anow The 
oicc Ihal will ha\c lo be 

ppheel Lo duiw it away 01 to Q \ 

told it fiom coming back is 
!iS7 whcic I is the length Ali 
Liiel S is the tension of the 
in face layer on each side of " 

he film pci unit of length 

Fi" 97 
Hie quantity S so defined s 

iH'asuics the surface tension of the liquid In chawing the loel 
iway tlnough any distance x m the dnection of the ariow the 
voik done is 26*^ Hence S also measuies the woik clone in 



344 THERMODYNAMICS [AFP 

forming a single suiface layer, per unit of aica of the layer; m 
othei woids, S measiues the potential cneigy thai is sloud m each 
unit of aiea of the free suiface of a liquid in con-sequence of its 
surface tension. 

It follows that the siufacc eneigy of a sphencal diop (that Is lo 
say the potential cneigy vuuch is clue to its siufacc tension) it, 
4>7T)*S wheie ? is the laclius oi the diop. 

The spherical foim winch a fiee diop assumes is Ihc Joim winch 
will make the suiface enemy (foi a given volume) ,i minimum. A 
chop resting on a suppoit takes such a foim as will make ils lolal 
potential eneigy a minimum, namely Ihc sum of Lhc cncigy of 
suiface tension and the eneigy of position which the chop has in 
consequence of the height of its centie oi i>ia\ily alnnc (he level 
ot the suppoit 

216 Need of a Nucleus Imagine a diopio beo apoialmi> nuclei 
conditions that keep its tempeiatuie constant Encit>\ lias lc> be 
supplied in piopoition to its loss of mass Lo pio\idc foi I he Lit cut 
heat of the vapoui that is foimed But the chop is losinn sui ffjce 
eneigy in consequence of its diminution ol suiface, and to sonic 
extent this reduction of suiface cnciyy supplies Ihc I.itenl Iic.it 
that is lequiied, only the lemamdei has Lo be supplied hoin oul- 
side the chop Consequently a chop is moie u-ad\ Lo (\aporalc 
than the same liquid in bulk, at the same tcnipciaLuie, and it will 
continue to e\apoiatemtoan atmosphcic \\lucli \\ould be sal 111*1 1 rd 
with respect to the same liquid in bulk Moicoxu, .is the 1 diop 
gets smaller and smallci (if we assume lhat the leduclion ol' si/c 
ma\ go on without altennq the natuie of smfaee (tiision), a sln^c 
Mould be icached when the loss of potential cneiuy due lo con- 
tiaction of the suiface would become sufficient to supply ,ill I lie 
latent heat of the vapour that is passing ofl In lh.it c \ciil, no 
heat Mould have to be supplied fiom outside Lo complete Lhc 
c\apoiation of the diop it would become mheieiiLly unstable and 
A\oulcl flash into \apour 

Foi the same icason a chop cannot foim excc-pl aiound a, miolc'us, 
and the laigei the nucleus the moie icadily iL loims When 
paiticles of dust aie piesent in expanding vapoui, the liisl cliops 
to be foimed use them loi nuclei, as was shown by Aitkcn (Art. 
79), and only a small amount of supeisatuialion occurs be! ore such 
diops begin to foim The cloud of paiticles obscived by Wilson 
when dust-fiee air containing Avatei -vapour is expanded enough to 



i] EFFECTS OF SURFACE TENSION 345 

cause an eight-fold supersatmation aie foimed aiound much 
s mallei nuclei which consist, piobably, of accidental co-aggiega- 
tions of the molecules of the gas itself, 01 of electrically charged 
molecules, such as aic always piesent in small numbeis It should 
be added lhat the presence of an electnc charge gieatly favouis 
condensation of the vapoui upon any nucleus As an electufied 
diop cvapoiatcs, the chaige lemains behind, the potential eneigy 
due toclcctniication theiefoie mcieases as the drop becomes smaller, 
foi I he cncigy due to a constant chaige vanes mveiselv as the ladius 
of I he sphcie that carncs it In this icspect the effect of an electnc 
chaige is opposite to Lhat of surface tension. Hence when a chop is 
chaigcd moic eneigy has to be supplied Iiom outside to make it 
cvapoiatc than Mould beicqunccl it it weie unchaiged An electn- 
cally chaigcd drop will theiefoie cvapoiate less icadily than an 
unchaiged diop of the same size, and maygiow laigei in an atmo- 
sphcie that is but little supei saturated 01 e\ en not supeisatuiated 
at all In \apoiu \\luch is slightly supeisatuiated it is found that 
an^ "loni/mg" action, such as that ot an electnc spaik, 01 of 
Uonlgcn lays 01 ot ultia\ lulet In-lit, \\hich sets fiee the ions 01 
pai ticks coineyuit" unbound electnc chaigcs, bungs about a cloud 
ot condeusalion, by ci eating fiesh nuclei, or )n stimulating the 
pouus ol existing nuclei thiongh causing them to acqimc an 
eketiic chaige ' 

217 Kelvin's Principle Confining oui attention, howc\ei, to 
diojjs which aic not cleclncally chaiged, \\eshall no\\ considei how, 
as a conseqiK IKC ol smlaee tension, the eqiulibiuim of a chop ot 
gi\iu si/e clc])tncls on I he slate oi supeisatiuation of the \apoui 
aionncl it Assume I hi liquid and the vapoui to be at the same 
[emperatmc Liquid with a Hat suiface is in cquihbiium with the 
\apom abo\e it when the vapoui is at the picssiuc. of saturation 
Ihc ir is lliui no tendency on the \\holc foi the liquid to e\apoiate 
OL loi the \apoiu to condense, any evapoiation that occuis being 
eviclJy balanced by an equal amount of condensation Liquid in 
Ihe lonn ol a small chop is, owing to its cmvccl suriace, in equi- 
hbnum with the suiioundmg vapoui only when the piessuie ot the 
\apoia suiiounding- it exceeds the nomial satmation piessiue by a 
(k'hnile amount, in othci woids, only when the vapour is supei- 
saluraUd The degree of supeisatiuation iieccssaiy foi cqui- 
libiium depends, on the curvature of the smface, and theiefoie on 

* Htei Ni ,1 J r riiomson, On the Condiution of Ekdncity though Oases, 
Cliap VI I, C T R WilTOii, Phil TUM* A.,\ol l'J2, 1839 



346 THERMODYNAMICS | MM- 

the smallness of the drop This principle Avas fust established by 
Loid Kelvin ! . It is of fundamental impoitance in explaining ihe 
retaided condensation of expanding steam 

We may apply Kelvin's gencial method as follows to find a 
iclation between P s the normal piessuic of satin ation (which is Ihc 
equilibrium vapom-piessuie ovei the flat sin face of a liquid | ) and 
P' the eqmhbimm vapoin-piessiuc OA ei a cmvcd MM lace, such as- 
the suiface of a i>mall diop. Take foi this put pose Ihc tuivcd 
suiface at A, fig. 9S, \\hich is foimcd by 
holding in the liquid a capillaiy tube of a """) 
matenal such that the liquid does not we I ^ 

it The column of liquid m the tube is 
accoichnglydepiessed thiough some distance 
h, and if the boie is small enough the ficc ft 

suiface at 4 is sensibly pait of a spheic -_= -^^-^=- 
Imagme the liquid to be contained in a \ 

closed vessel, and that the space C above I 

it contains nothing but the Aapom of the I 

liquid Let all be at one tempeiatuic T. h 

The "tthole system is in equilibiium 0\ci ' 

the fiat suiface at B theie is Aapom \\liost 1 ' 

pressuie is P s ovei the cun ed suilaec at , 

A theie is \ r apoui of a highci piessuic P' . ^_ 

The difference P' P s is equal to the 

weight of the column of \apom in the lube 

(pei unit aiea of cioss section) fiom the 

level of A to the level of B Let a be the 

weight of unit volume of the A apoui 11 ,, it ^ 

this weie constant, the \\ eight of Ihe column 

of \apom in the tube (per unit aiea of section) would be simply 

crh But a depends upon the piessnie P, it is equal lo I//" and 

may theiefoie be wntten p 

a = KT> 

if we take the equation PV= RT to apply. The chifcicncc in Ihe 
t^o vapoui piessuies is p> _ p _ r^^ 

mtegiated between the level at B and the level at A 

Compaie next the hydiostatic piessuies within the liquid just 

h PIOL Roy So( Echn ^ol \ar, 1870, Populm lect'ne* andAddn sac*, vo] 1,11 d I 
\ Namelj', the satiu f ifcioa pressure tor anj assigned tcmpoiatuio as given in tables 
of Ihe piopeilies of satiuated steam 



il EFFECTS OF SURFACE TENSION 317 

undei the surface at B and iusL nuclei the smface at A. Just under 
the flat sin face at B the hydiostatic pressure in the liquid is equal 
to the piessure of the vapour ovei the suiface; il is theiefoie equal 
to P s . As we go down Lhiough the liquid to the level of A, the 
hydiostatic picssiuc inci eases by the amount ph, where p i^ the 
weight of unit volume of the liquid Theiefoie just undei the 
cuived suiface at A its value is P^ + ph. 

But we may also calculate the hydrostatic piessuie under the 
cuived suiface at A in anothei way The Lop of the liquid column 
at A, which has a suiface layei m tension, may be tieated as a 
segment of a spheie of laclius ;. lib surface la\cr foims a cap 
whose suiface tension S causes it to pi ess down upon I he liquid 
belo\\ with a pressmc p such that m-p = 277/5". That this is so 
will be seen at once bv considmng the cquihbiium ol a complete 
hcnnspheie of the t>amc cunalurc and with the same suiface 
tension. Round the cuciimfeicnce (2n)) of the hoiizoutal plane 
foiming the l)asc of such a hennspheic time \vould be a \eitical 
foice 27T/.S' balancing the icsiiltaut foice due to the piessuie p 
acting on the aica of the base, 7r> 2 . Hence 

2,9 
P- , , 

and the hydiostatic piessuie pi^t undei Lhe cm\ r ed suiface is 

theiefoie equal to 9$ 

P' + 

/ 

Equating Ihc h\o cxpiessions for this h\ dm^lalic piessuie, we have 

1 >,V 
P' + j - J\+pli, 

01 ' 2<S = p h-(P' - 1\}. 

) r 

Hence, since P' P s = $adh, 

2S 

= ph $adh ~ \(p a) (III. 

And since dP = adh, 

2S ( p 'p - a ._ [ p 'p |D . 

- = - - dP = - dP veiy neaily, 
? 'r* 'PR " 

because a is small compared with p. 

On substituting P/RT for a this appi oximation gives 
2S l p 'dP P' 



* 6 

P' 

\fT _ 

c P 



348 THERMODYNAMICS [VPP 

This applies to any liquid suifacc whose radius of cui\ al lire is i It 
therefoic expiesses the iclation of the piessiue P' m tlic supcr- 
saturated vapour aiound aspheiical diop of ladius > to Lhc noimal 
pressine of saturation P s (over a flat sniface) Toi the same Lempci a- 
ture, when the diop is in equilibrium, in the sense that" iL is ncithci 
giowing by condensation nor shimking by evapoialion Tlic 
expression shows ho\\ the dcoue of supeisatuiation P'H\ ncccssai y 
foi the equilibiium of the diop mcieascs when the size of the diop 
is i educed For a diop of gi\ en ladnis anv mcicase of P abo\ e the 
value so calculated would cause Lhc diop to giou The expression 
also shows what is the least size oi diop thai can exist in an 
atmospheie with a gn en dcgiec of supcisatui.nl ion an> diop foi 
which i is smaller A\ould chsappeai by c\apoiation 

Ib is only when the diop is very small that the excess of /" 
ovei P s is at all considciable This is best sho\\n by numtiical 
examples If \\e take watei-vapoui at 10 C 01 '28-3 absolute, and 
usec g s units, RT (which is ticaled as equal to PT) is 1 30 H)' 1 
The sui face tension of watei at lhatlcmpcialuie is about 70 dynes 
pei hneai centimetie,and/-jis 1 qiammcpci cubiccentimclic Hence 

, P' 2s 76 I 01 

[(")(-) == 

010 P, 130 )- 10 s / 2303' D ' 

where D is the chametei of the diop in milhonths of a millimctte 
The foimula accoidmgh gives these icsults 

Ratio of Vtipoiu pussiiii 1 

Diameter uf diop in equilibiium \vitli the 

in inilhonllis of clif>i) to nunn.il -,alui.i 

<i nulbmetic tmn pio-,siiic fni iho^aiiu 

ttinpi i.itino (P /Pj 
100 1 02 

r )0 1 O r > 

10 1 2b 

r ) 1 f,9 

2 .32 

1 102 

This means, for instance, that a diop of watci two milhonths of a 
millimetie in diameter vill giow if the latio of supcisatuialion in 
the vapour around it is gioatci than 3 2, buL will evapoialc iJ 
that latio is less. Hence when the idtio of siipcrsatmalion is 3 2, 
drops uill not foim unless theie aie nuclei picscnl which aic al 
least big enough to be equivalent to sphcics with a diameter of two 
milhonths of a, millimetie. 

In Wilson's experiments a cloud of mist was produced when the 

supeisatuiation was S, which coiicsponds, by the foimula, to a 

* To comett from Napienan to common 



ij EFFECTS OF SURFACE TENS JON 349 

diameter of about 1 1 milhonths of amilhmetie On the assumption 
that the foimula may still be applied to such small nuclei, it might 
be mfeirecl, if theie were no loiuzatjon, that water-vapour contains 
many nuclei of that oidci of magnitude, which may be pans or 
small gioups of molecules co-aggregated by chance encounteis. 

IL will be obvious fiom Kelvin's punciplc that a chop of water 
cannot continue to exist in an atmosphcie of sal mated vapour 
When the diop and the almosphei'e aie at the same temperature, 
the diop can exist only if the atmospheic aiound it is super- 
saturated. Foi any given degiec of siipeisatuiation theie is a 
value of / (determined by the foimula) such that a chop of smaller 
radius will evaporate and a drop of laigei lachus will glow Thus 
the bigger drops in a cloud Mill tend to giow at the expense of the 
smaller chops. 

218 Ebullition Similar considerations go\ern the foi matron 
of bubbles in a boiling liquid We may tieat any small bubble as a 
sphencal space of lachus >, containing gas, bounded by a spherical 
envelope in which theie is surlace tension Outside of that is the 
liquid, at a pressure P In consequence ol the surface tension in the 
envelope, the pressure inside Ihc bubble P t must exceed P bv. the 
amount 2S/), \vhcre A' is the surface tension in Ihe boundarj surface 

of Ihc bubble, making <?V 

I* _ p - 
1 / 

\Vhen / is very small this implies a gieat excess ol pressure within 
the bubble If no pai tides of air or othei nuclei vuic present to 
start the formation ol bubbles, boiling \vould not begin until the 
temperature \scie raised much above the point coi responding to 
the external picssuie, and would occui \\itli almost explosive 
violence Once formed a bubble would be highlv unstable, for as 
the radius increases the tension ol' the envelope becomes less and 
lest, able to balance Ihc excess of pressure within it This happens, 
to some extent, \\hcn \\ater is boiled after being ficed of an in 
solution it is then said to boil with bumping 

It follows that a pure liquid may be superheated, that is to say, 
] a i seel above Ihc temperature of saturation corresponding to the 
actual pressure. Tins is an example of a metastablc state like the 
state that is produced when a vapour is supercooled without 
condensing, or when a liquid is supercooled wrthout solidifying 
Water at atmospheric pressure may be heated to 180 C or nrore 
when it has been freed of air and when it is kept from contact with 



350 THERMODYNAMICS [AIT i 

the sides of the vessel by supporting it m oil oi its own dcusiLy, so 
that the Avatcr takes Lhe ibim of a large globule immciscd in oil 

In the ordinal y piocess of boiling, a bubble contains in general 
some air 01 other gas besides the vapour of the liquid iLsell. With- 
out gas in it, the bubble could not exist in stable equilibimm 
With gas in it, the bubble will be m stable eqmlibiiuin when the 
paitial pressiue due to the gas piOMdes the necessary excess of 
the whole internal piessure P i ovei the external piessine P Any 
i eduction of the bubble's size would then raise Lhc piessine of I he 
gas more than enough to balance the increase of 26'/j. Let P v be 
the vapoiu -pressure inside the bubble If we assume that Lhe 
external pressure and temperature remain constant, the pailial 
piebsme due to the gas ma}' be expressed as a/> 3 whcie a is a con- 
stant. Then P l = P v -I <7/> 3 , and the equation 



deteimmes the \alue of / at which the bubble is in equilibrium 
The quantity P v P is the excess of the A apoui-picssiuc in the 
bubble o\er the pressure in the liquid Dillcientiatmg this with 
respect to ;, to find the limiting condition loi stabihh , \\e ha\ e 



, 

A\ lien , - , 
i i 3 3 / 

and therefore when P u - P = 

ol 

Hence foi stability P V P must be less than 4>S/'3t. Tins 
means that \\hcn a liquid containing bubbles of ladius > it. 
heated, the temperature will use until the \apour-piessurc within 

the bubbles exceeds the pressure in the liquid by the amount t , 

but when that point is leached Lhe bubbles will become unstable 
and ebullition, will begin Callendai ' calculates on this b.isis IhaL 
Avith bubbles one millimetre in diamclci water (under one 
atmospheie) will boil at a tempciature of 100 05 C , and that 
to pioduce 10 of supeiheaL the diameter 1 of the bubbles must not 
be more than about - ^ mm. 

* Enc Bid , AiLiclc 'Yapouzation " 



APPENDIX It 

MOLECULAR THEORY OF GASES 

219 Pressure due to Molecular Impacts Accoi cling to the 
moleciilui thcoiy, a gas consists of a \ ei y large numbei of paiticles 
called molecules men ing with great velocity Each molecule moves 
freely, with umfoim velocity m a shaight line, except when it 
cncounleis anolhci molecule 01 the wall of the containing vessel 
In an cncountci the velocity changes in chiection, and gcncially 
in amount, but theie is no dissipation of encigy, the mole- 
cules bcha\ e like pcifecth elasLic bodies As a icsult of 
manv cncountcis, a stable chstiibution of speed among the mole- 
cuks is established but the speed of any one molecule is being 
cotislaullv changed, b^ its cncountcis, within vuy \\idc limits. 
The length ol llu 1'ieo path, \\huh il ha\ciscs between one en- 
couulci ,ind Hie nc\l, is also quite nicgulai The a\ciagc of that 
leniilh, 01 Mhal is called Ihe ' mean fice palh," is \ en long coni- 
paud \\ith the dune usions of the molecule, ilsc'lf This chaiacLeiislic 
distinguishes a i>as horn a lujuid Jn a, gas the a\ e.iagc Lime dining 
uluch a molecule is moving in ils lice path is \ei\ laige compaicd 
\\ith Mie dme of an cncoimlci ]3\ r the lime oJ an encounter is 
mcanl Ihi lime dining which the molecule is cithci in conlact 
wilh anothci, 01 so ncai it that thcie is a sensible (nice acting 
be [\\een them "When a gas is coinpu'ssed, Die mean liee path 
is icduccd, .u id the eneounteis become nioic lietjuctit bct\\cen 
one molecule and anolhei and also be I ween Ihe molecules and 
[he 1 \\alls ol the \csscl When a gas is healed the speed with 
which Ihe molecules mo\e is incieased, we shall see, immediately 
thai then a\ eiage kinclie enci<>^ is piopoilioiul to the leinpciatmc 
The molcculai thcoiy is now well established theie is conclusive 
evidence that actual gases do consist ot pailiclcs moMiig in the 
mannci which the Ihcoiy piescubcs 

The ]}icssmc of the gas, Ihat is lo say, the picssure which the 
gas excits on every unit of smface of the containing vessel, is clue 
cntiiely to the blows of the molecules upon the surface the mo- 
mentum given to the sinface by their blows, per unit of area and 
pei unit of time, measuies the pressure m kinetic units. 



352 THERMODYNAMICS [M>P 

In any gas thai is chemically homogeneous, all I he molecules 
have the same mass Call that mass m. Lei N be the numbei of 
molecules piesent m unit of \olume ol' tho gas in any actual sLutc 
as to piessme and tcmpeiatuie Then inN icpiesents Lhe densiLy, 
namely, the whole mass per unit ot volume, and V, Die \oluuio 
pci unit of mass, is equal to 1/inN 

Befoie piocccding to considei the picssinc caused bj molecular 
blows, we shall make the following postulates, 

(1) That the molecules aic pcil'cctly lice except dining cn- 
counteis, and thciefoic move m stiaight lines with uinloim 
A elocitj , fiom one encounter to the next, 

(2) That the time dining which an cncountei Lists is negligibly 
small in compaiison with the time dining which the molecule is 
fiee, 

(3) That the dimensions of a molecule aie negligibly small in 
compaiison Auth the fiee path 

These tlnee postulates, aic equivalent lo assuming that Lhe gas 
is pei feet in the sense of Ait 18 Tile's aic not shicllv ti uc of any 
ical gas, but we shall assume them to be line m what immediately 
iollows, and shall theiebj deduce Uom the molcculai theoiy a 
lesult which coiiesponds to the ideal loimula l*V ^ RT 

Suppose the gas to be in ccmilibuum m a vessel at usl, anel lei 
Ihe velocity v ol any molecule be lesolxcel mlo icclangiil.n com- 
ponents v^, v u and v,, along tluec fixe el <i\cs 

Considei the piessme due to molecuLu blows upon a containing 
wall, of aiea S, fanning a plane suiface al nglil angles to the 
dnection of x The conliibution which any molecule makes to the 
piessme on that wall is due cntncly to Ihe component \elocily 
v, nothing is contnbuted by the components v a 01 v. Any molecule 
which stnkcs the wall has the noimal component ol Us velocils 
icvciscd bj the collision lie nee the momenlum due to Ihe blow 
is 2i>iVj_ \\hcic i\ is Ihe noimal component of I lie \eloeily anel in is 
the mass of the molecule. 

Considei next how to expiess the sum ol the e fleets ol such 
blows in a given time Foi tins puiposc we may think ol the mole- 
cules as divided into gioups aceoieling to Ihcn \elocihes at any 
instant. Let n be the number, m unit volume ol the gas, whose 
^-component of velocity, v^, has the same numeiical value Since 
the numbei of molecules is very gicat, we may take the munbci 
to be the same m one cubic inch (say) as in anothci. Thcic will of 
comse be very many such groups, each with a diffeient \aluc of 



c] MOLECULAR THEORY OF GASES 353 

t Think, in the (list place, only of those in the gioup n. Half of 
he whole numbei of molecules in the gioup aie moving towaids 
', the other half arc moving away fiom it At any instant of time 
lieic Avill thcicfoie be within a small distance Sa' of the smface S, 
nd moving towaicls iL with component Aclocity i\, a inimbei of 
lolccules of that gioup equal to }n$Sd\ A molecule distant S,r 
om >, and having a component velocity v^ towaids >S', would 

^\ 

idch S in a time St --- J piovided it did not encounter any othci 

lolcculc on its way Ilence the number of blows dehveied to S 
y molecules of that gioup, in the time Sf, would (on the same 
LO\ISO) be equal to the numbei of such molecules a,s onginally 
y within a distance S,t, namely the number \nS8x 

Hence also I he momentum due to the blows on the aica S in the 
me 8t would be equal to \nS5iU ,c SiflVj , which becomes, pei unit 

aiea and pci unit of tune, 

& 

nitn\ ~. -- nmi\-, 
of 

BK 

v ^Wt 

This is UK momentum contiibutul by one yioup only The 
essiue P is nude up of Ihe MUH of llic quaulitics of momentum 
nlnbiited l)^ all the yioups, hence 



UK N is ,is btlou llu \\holc numbei of molecules pi.i unit of 
liime, and i 1 ,- is Ihc 1 t i\ i I<IL>C of v,~ loi all Ihe molceulcs 
Now I lie \eloe ily i' of an\ molecule is icLited to its components 
the equation ..> _ .. j , i , , 2 

I - ( , I" l'^ I L y 

i nce, it u r c \\tiley-foi the a\ ciagc value of i "loi all the molecules, 

v 2=s w, 3 + v y -+ v, 2 = 3v, 2 , 

cc the motions take place equally in all diicclions 
The square loot of v- is called the " velocity of mean squaic " II 
lot the same thing as the average velocity, but is the velocity a 
'lecule would have whose kinetic eneigy is equal to the average 
ictic energy of all the molecules 
The expression foi P may therefore be wiitten 

P = 



354 THERMODYNAMICS |APP. 

FmHiei, since mN is the quantity of gas in unit" volume, 01 1/F, 
where V is (as usual) the volume of unit mass, this gives 

PV = \v*. 

In obtaining this result we made (in oidei to simplify I lie aigu- 
menl) a proviso that each molecule of a pnilic-iilai gioup, lying 
initially within the distance ,r of tlic wall, shuck I he wall without 
encountering othci molecules on the way. This is not line, hut any 
encouulci on the Avay docs not alfcct Lhc final lesnll in a gas to 
uhich the thiee postulates apply. For in any cneountei, some 
momentum, peipcndicnLu 1 to the wall, is simply tiausfc'i ted to 
anothei molecule, and i caches the wall \\ilhoul loss The molecule 
which takes it up has to tiavcl the full icmamdei of the distance 
in the clucction of .r, neithci moie noi less, since I he dimensions of 
the molecules aic negligibly small (Poslulale ,'*), and no him is 
lost in the encountci (Postulate 2) Hi nee Hie geneial usull ot 
the cnconntcis is not to altci the amount <>l monunlum \\lnch 
leaches the wall in any given time, and the conclusion u mains 
valid that PV =- 'w 3 

C'ompaimg this \\iih Lhe pcifccl-yas equation 

PV= RT 

we t,ec that u 2 is piopoitional to I he absoluk U mpc lalmc. , and 
consequently the a\ciagc kinetic cncigy \\lneh UK mohciilcs 
possess m Mituc of then \clocily of hanslahon is pmpoi honal lo 
the absolute tcmpciatmc We shall call then entity ol haiislalion 
E' , they may, in addition, ha\e cncigy of olhu kinds 

The encigy of translation of the molecule s 7i" is ei|iial lo \c~ pel 
unit mass of the gas. Hence by the moleeuku lluoiy 

PV _ 2 7," 
L V iJ'j , 

and the picssinc is equal Lo two-thuds ol the cm my of tiauslalion, 
pei unit volume of the gas. 

It may be noted in passing thai I he moleeulai llieoiy explains 
why a gas is heated by compicssion Think of I he gas .is contained 
in a cjlmdci, and being compicsscd by the pushing in ol a piston 
Then any molecule which stukcs Ihe piston recoils with an increased 
velocity because it has struck a body that is ad\ ancing towards 
it. The component \elocity v y normal to the piston is not simply 
leversed by the blow, but is increased by an amount 2z/, where v r 
is the velocity with which the piston is moving when the molecule 



c] MOLECULAR THEORY OF GASES 355 

tnkcs it, for the quantity which is ic^evsccl is the iclative 
cloc'ity v^ + v'. The icsult is that the motion of the piston in 
omprcssmg the gas augments the a\ eiage velocity of the molecules, 
nd consequently mci cases v' 2 , on which the temperature depends. 

220 Boyle's, Avogadro's, and Dalton's Laws These laws 
)llow from the molecnUi Iheorv, foi gases I hat obey the thice 
Obliilates Keeping v 2 constant, we have the law ol Bo} r lc, 
'V = constant, since PV }v* 

If Iheic ,ue two gases at the same piessnic, since 7* ]//>A 7 y 2 

1 CMchl "'i^Vi 2 ' - M 2 N 2 v 2 * 

a\\u'Il has sliown that if h\o gases aie at the same Lem- 
lalme, the average kinetic cneigy of a molecule is the same 

both, 01 -7 , 

ninY =- w 2 zv 

encc li they aie al Hie saint piessuie and the saint Lcmpcraliue 

A\ = A',, 

<il is lo sav, Ihe nuinbei ol inoleeuk s in unit \olinnc is the same 
i bolh, \\lnch is A\oi>adm\ Law IL lollems lhat the density, 01 
rissol unit \ ohiinr 1 , difk i s in I he t \\o Base's in the laliooi I lie masses 

then molecules, 01, in olhu \\oids, the elensit\ is piopoilional 

Hie inoIeenLii \\eiyhl (Vit 15S) : 
Again, the nioli enl.n theoi\ sho\\s lhal in a nnvtiiie of t\\o 01 

>ie gases, e.teli ol uliich obe\ s (he thie'< k postulates, 

/' =-- \niiN f\* -I .'/ ,A r ^ -I- etc 

ol In t \\oids, I lu ]iai lidl piessuie due to each eonslil ne'iit of Ihc 
\luie is the sanu .is il nonld be il the othei conslitucnls weic 
t theie This is in agieeiuent \\ilh Dalton's Law (Art (}'2). 

Tin.' unmix i j\ r ol niolmilis pci culm if nliiwf 10, v\liic-li H Ihc s.iino Fot 
Moiilr fiii^fH n(. llu Sdiiif lnii|n inliiin ami picsHiuc', n ahnuL 27 5 ^ 10 lft foi any 
<il ()" mid .1 pus-'UK ul one uimosplioie (KOO Jc'.ins 1 Di/mumml Tlnoi i/ of 

('ft, p S) Tlicn ivoi'iffo (iHliuuo iipail, A\!UC!I ia |7 rr r IH tlioicfoio about one 



e niilhoiiHis of a coiitimelio The nuinboi of molcciil<"( pci inol is 22100 N or 
. x 10-' 

ho maRg 7>i of a nioloculo m any gns ma)' bo found by dividing fcho density 
by N Since the density of oxygen is 0011-29 ginmmo per c c tho mass of 
xys^n molecule is about 52 x 10-- 1 giamines The mass of a hydroprrn molecule 
ie-si\lecuth of tliat, 01 about 3 3 x 10~ 21 giamines, tlie latio of the molecular 
lits being 2 to 32 



356 THERMODYNAMICS [ u'i' 

221. Perfect and Imperfect Gases Thus I ho mola-nlai llic.i\, 
for gases which satisfy the tlucc poslnlalcs, gives lesiills idcnlic.i! 
with those we already know as laws of ideal pi i (<'<! i>.si s 

In a ical gas the postulates do not si i id Iy hold. Tlu si/i ol (In 
molecules is not negligible, and in any cnc-onnli'i I lu'ie is .in .ippn - 
ciable time dining which the molecules conevi m (I < vi I lmr(s on 
one anothei Theie may even bo lempoiaiy p.m m<> <>i cn-,inir.i- 
tion on the pait ol some molecules It is ink ie-,1 mt> In i IKJIIIIC, in 
aoeneial way, how these dcpai lint's lioin I he idi.il coiidihons all-l 
the calculation of the picssuic 

For this pin pose, considei Ihe simple case in winch one ol ,i 
gionp of molecules, advancing louauls lhc wall, meclsa molidili, 
initially at lest, to which it pisses on UK whoh ol ils inonu nluni, 
and the othei molecule then completes UK IOIIIMIN and ddmis 
the blow If theie weic no loss ol Lnnc in UK uicoimlu, and il I IK 
second molecule could be ie<>aided as havdlnm o\ ci < \adly llu 
lemamclei ol the distance, Ihe uile .il \\ln<li I lit' \\ill i-<i\<s 
momentum uonlcl be exacll> I lie same .is il I he c IK onnh i li.id nol 
taken place But if theie wcie loss ol lime in ,IIIN < ..... nnhi -IK h 
for example, as would ocein il Ihe hvo colliding niol(<iil(s mo\(l 
togethci foi any appiccuibk lime', uilh Hun \<lo(il\ xdiuid 
below that of the molecule \\hich \\.is onnin.ill\ imuin^ linn 
the late at which the \\,dl uti'i\ ts nionu nl inn \\ onlil l>< K diu \ d 
with the icsult of leducniii /' On UK olliu hand, il I In moliinhs 
have a finite size, so IhaL Ihe OIK \\huh \\as nnli.dh .il u^l had 
less distance to tia\cl in compk lino Hie IOIIIIIM , llu ial< ,il \\huli 
the impacts succeed one anolhei on Ihe \\all \\oiild Ix in<'i(as<d 
with the result of meicasmo /' 

This indicates thai Ihe pussine in a i< al i^as \\ ill dilh i lioin I IK 
ideal piessuie, which is ni\en by Ihe e(|iialion 1*1 ',,>' l>\ l\\o 
small teims, one posili\e, de peiulmg on Iliesi/t ol llu moh <'iiU s, 
and one negatne, depending on Ihcn eolusion. Such, in ilhrl, 
is the kind ol moelificati on \vlnch finds t \piession in chaiaclu ist u- 
equations like those ol Van dei \Vaals, C'lausuis, 01 C.illi ndai 

222 Calculation of the Velocity of Mean Square Taking, loi 
any gas that may be tieated as sensibly pc ileel, I he i iinalion 



it is easy to calculate Ihe value of Ihe velocily ol mean sqimii' ,' 
when we know the density ol Ihe gas al a yive'ii pressmr The 



ill MOLECULAR THEORY OF GASES 357 

pioclucl mN is the density, and we do noL need to know m or N 
scpuiatcly Lo find v In oxygen, for example, at C , the density 
is 001 1.20 giammc pci c c , when the pressme is one atmosphere, 
or 1 0133 x 10 G dynes pei sq. cm (Ait. 12) Hence in oxygen at 

, , , /3X l~0133x ]0 G , 

standaid LemperaUne andpicssmc, ens /* / --- nT^i --- ' ec l ual 

V W \ I 'J A. ^l l Lt J 

Lo JG1 mctics pci second Sinnlaily in niLiogcn it is 193 melrcs per 
second, and in hydiogen 1S30 metres per second 

223. Internal Energy and Specific Heat. Consider next the 
beaimg of Lhc mokcnlai theory on the internal cneig} and specific 
heals ol a ^as We have seen that, in an ideal gas, 



wheic E' is Ihc encigy of lianslalion of tlic molecules, or ]>nNv z . 
This ma> be will ten 

KT = \E' 01 E' = \RT 

E' is llicitJoic piojioi lional lo the Icmpciahne Now E' may 01 

ma\ not be Hie whole inleinal uitioy, E, which the nas acquius 

u lie ii il is lic'.iicd It will be (he whole li, when Ihc "as is heated, 

he molecule's can onl\ lake up enei^\ of tianslalion, and cannot 

,ikc up (ii<io\ r ol lolalion 01 tncigv of \ibialion (\it 171). 

iupposi , loi instance, llial each molecule beluncd like a pci feet ly 

iiuoolh MUM! hilluid ball, 01 like ,i massix e point \\iLh no ajipicci- 

il)le moment ol meilia about an\ line passing lhioiii>h it In thai 

as( il could not ha\e aiu r encii|\ ol vibialiou, noi ,if<]uiie' nnv 

IK 1 1>\ ol i dl at ion in I he couisc ol ils e'licoiuile is \\ ilh olhei molc- 

uhs, ,iu(l llic oul\ kind ol e'oiiummie'.iltle kmelie eiuigy \\oulel 

je (iiein\ ol lianslalion \Ve should Ihen find /',' = /',", <inel con- 

(([iKiilly E - \RT 

\\'\\( n a gas ol this kind is heated, we' shoulel theiefoic have 

dK- \KdT 

iul in anv gas (icgaided as ])eil'ecl ) 

(IE - A',//'/ 1 anel A',, - K, -|- R 

lence I'oi a ^><is ^ hose molecules have energy oC lianslaLion only 
K = Ut K - r 'K 

JV u J > l> i ' 

K 5 

ud y or -_^ = - 01 1 007 

A .3 

'his value of y would not apphy if E' wcic only a pait of E But 
is found that in a monalomic gas, such ds argon, or hclmm, or the 



358 THERMODYNAMICS [APP 

vapour of meicuiy, the value of y is in lad cc[ii,'il lo "I 007 01 very 
near it The infeience is that in a monatomio gas, Ihc stmcluic of 
the molecule is such that substantially all its communicable 
energy consists of energy of translation 

In any gas each molecule possesses llucc dcgices ol freedom of 
translation, namely, freedom to move along cuch ol Lhiec inde- 
pendent axes. Since E' = ^RT, each ckgice of freedom ol lians- 
lation accounts foi a quanlity of kinetic energy equal to \RT 
This is tine whatever be the numbci ol aloms in I lie molecule, 
and whether or no the molecules have othei eueigv besides eneigy 
of tianslation. 

Consider next a diatomic gas, each molecule of \\lueh consists 
of two atoms. According to modem views 1 an atom is a eompUx 
system made up of a minute positn eh chaiged ccnhal nucleus in 
which the mass of the atom is almost all conccnliatcd, with 
electrically ncgati\c particles called elections elislribuled .uound 
it, at distances which arc large compared \\ilh tlu dune nsions of 
the nucleus [. The sliuctuic of the ,ilom and lhc naluic e>l Ihe 
forces between one atom and another m tlie 1 mole'eule au slill 
uncertain, but lor our present purpose it will suffice lo pie-hue 
an atom as a niassn e point, smiemndeel b\ a masshss quasi-claslie 
lender clue to forces which keep olhei aloms ,il a dislance Uuehi 
normal conditions a diatomic molecule is equi\al<ul, as regauls 
inertia, to two masses helel some dislance apail dynamically il 
may be compared to a dumb-LJI, a mou- e\ael compai ison would 
be to a light stick capable ol some elastic cxluision and can \mg 
a hcavj ball at each end Consideied <is a ngid body il has h\e 
effective degiees of freedom elfectue as icgauls UK slonug ,ind 
communicalion of kinetic cue igy name l\ r , I luce ol hanslal ion and 
two of rotation [ The two cfl'cclu e dcgic'es ol htedomol lolalioti 
arc about axes in a plane pcipcndieulai lo Uu x line joining Ihe 1 I wo 
atoms about that hue Usell, Ihe system has no elk el i\ e moment ol 

" Su ttiilliitlMiil, Pli 1 1 l/V//,M.i\ l')ll Bnln l >: nl Mini .Julv, St'pi and 
No\ l ( )U,,f J Tlioiuson Phil MIKI A|inl, !')!<) 

"I la an clcctiicilly iiculiiil alum lln< i)osi(ivc < l(u tin il v in (In inn IMI-J it (<|ii,il 
to the npy;,ilLvcclec!Liiut3 in (he I'lc'ctious Konmval ol ono m IIKIKMI! (In c'ldlnuiH 
would llioicfoic lcn\o tin. atom us ,i wlifilo |IOHI| ncily < lniij, f ( d I IIIH hiip|i( us wlion 
d ^is is ' iiuiiZL'd " 

\ AiiLC ii^id hcid\ luia si\ doniccs of IKUM|OIH il ( uu nio\c pniallit! lo itsell 
alony thioa iiidcpendenl a\oa, mid it mil iohil(^ about lhcs( avi's 'Vny poHsihh 
mo\emcnt i^ marie up of Ihcso MI\ components In a (lialoinn inoh uiln one of I lies 
degieos of freedom of lotation is inelloclivo us it)u;ai(N thu coiiiiunnicadoii of 
energy from one tuoleciile to anotliei in an cncountci 



] MOLECULAR THEORY OF GASES 359 

cilia Under these conditions it can be shown lhat Lhc ultimate 
suit of collisions is that the kinetic energy becomes equally 
laied by each of the five degiees of freedom The energy of 
anslation E' is equal to }RT, and each degree of freedom of 
anslation accounts foi an amount of eneigy equal to }RT It 
Hows that each of the two degiees of freedom of i otation accounts 
i addition foi }RT, and that the energy of translation and rotation 
>gclhci amounts to r [RT Hence if theic weic no sensible cncigy of 
ibiation as well, we should have the whole eneigy E = ~RT and 

K j, = r ',R, K v = 7 7 /t\ and y = | 01 14 

Now in most diatomic gases, such as oxygen (O,) 3 mtiogcn (N 2 ), 
u, hydiogcn (II 2 ), nitnc oxide (NO), or caibomc oxide (CO), it 

in tact found that y is equal, very ncaily, to 1 -i at oidmaiy tem- 
ciatuies, and the mfcicncc is Lhat the stiuctuie of their molecules 

such as to give n\ c effective degiccs oi freedom, namely the fi\ e 
ml ha\e just been dcscnbcd, and that then molecules do not, at 
iduui v tcmpcialuics, hold anv consideiablc amount of commum- 
iblc ciK'igy m any othci fomi than as cncigv of tianslation and 
iK'igV ol jotalion But when such gases ate stiougl\ heated we 
no\\ lhal the specific heal mcicascs andy is i educed This means 
lat cneij>\ of Mbialion is then developed, which at high tcmpeia- 
IKS becomes an impoilanl pai t of the whole encigv 

In tiiatomic gases it mav be coii|cctuud that the iluce 
loins of anv molecule gioup lliemschcs not in one sliaight 
uc \\lueh would be an uiislable aiiangcmonl but so that 
le massive cuihcs he at Hie- cuineis of a tiiangle Sinulaily 
'ht'ii Iheie 1 aie moie lhau Iluce atoms m the molecule, they 
'ill place lhemsel\cs with llu u massive ecnhcs at the coincis 
I a polyhediou In any such Uiangulai 01 polyhcdial stiuetuie, 
onsidoied as a ngid system, thcic aie si\ effective degiees of 
eedom, nainelv thiec of lotation as well as thiee ol tians- 
ilion, loi theie is a liuitc moment of meili.i abonl anv axis, 
nd I he siiucture is such that Ihc molecule can be sel spinmug 
bout any axis by eneounteis wilh othei molecules As an ultimate 
ustilt of manv such encountcib, it may be shown lhal each of the 
hice degiees of freedom of lolation hikes up a shaic ol Ihc kinetic 
nergy equal to that of each of the tluee degiees of freedom of 
uuiblation, namely, \RT, and consequ cully that 1hc six degiees 
ogcthci account foi a total of SRT That is the energy which 
he molecules possess in virtue of their movements as rigid 



360 THERMODYNAMICS [ \pp 

stiuclmes If tlieic weie no othei way in which Ihcy could take 
up energy when the gas is heated, we should consequently find, 
in a tuatomic 01 polyatomic gas, 

K = 37?, K p = itt, and y = \ or 1 333 

The actual value of y, as expei imentally mcasuicd, in Lhc In- 
atomic gases C0 2 andII 2 0, is lathei less than Lhis, and in gases of 
more complex consLitution it is generally a good deal less IL is 
also found that the specific heals aic gicaler lhan 3/i* and \>R The 
mfeience is that m such gases the molecule geneially takes up a 
consideiable amount of cnerg} T oi vibiation m addition lo its cneigy 
of tianslation and lotation It appeals Lh.it a complex molecule 
can absoib energy not onl} by moving as a ngid body but by 
internal vibratory movements which ausc Ihiongh quasi-elastic 
defoimation of its own stiuctiire (Compaie Ai I 17 J ) 

The mam parl ol this encigy of vibiation piobably consists of 
to and fio jno\ cmcnts on the pait of the massi\e ceulies of Ihe 
linked atoms It is obvious that such a motion might ocem in 
any molecule that is made up ol moie than OIK atom Tin illect 
m a complex molecule is such as uould occm il the lines (oniing 
the massn e centies of the constitucnl alonis behaved like still 
spiings Thus m a diatomic molecule tvc might Hunk of the 
"dumb-bell" as haMng an clastic shank which allowed the distance 
between the h\o masses to ^alv The I'.icl th.il in a diatomic gas 
at oiehnaiv tempeiatmes the obscned specific heals arc <ippio\i- 
mately : lR and y?, and y is appioximaleh ] I, shous, Iie)\\c\ei, 
that the diatomic molecule then bcha\ es like a dumb-bell \\ilh a 
neaily mextensiblc shank BuL when Ihc tc'inpcialme is high, the 
vibiatoiy motion becomes iclalively moic nnpoit.inl, and il 
accounts loi an appicciablc pait of the whole encigy, c\ en in a 
diatomic molecule, and still moic in a liiatonuc 01 polyatomic 
molecule To this AVC must ascnbe the progiessixc mciease in 
specific heat, and the fall in y, which aic- obscived when any gas 
is heated that has t\vo 01 more alonis in the mokcule 

In a mona tonne gas Lhcie is no possibility of llns kind of vibui- 
tory motion, and theie is no cxpciimcnlal evidence of any change 
of specific heat vtith tempcratuic. The eiuigy depends only on 
motion of tianslation, and when the gas is heated its eneigy 
mci cases m simple piopoition to the Icmpeialuic I3ul when 
diatomic, tuatomic, 01 polyatomic gases aic shonglv heated, the 
eneigy mcieascs m a moie lapid latio than the lcm])eiature. This 



ij MOLECULAR THEORY OF GASES ,361 

means thai the ratio of the total energy E to the energy ol trans- 
alion E' is not constant. 
In any gas that satisfies Lhe equation PV = RT, 

-, R 
y=l+ r . 

JX - U 

[f the total energy E presented a constant ratio to E', the specific 
icat would be constant, and in that case we should have y> constant 
ind equal to 1 + jE/E', since E, icckoned fiom the absolute zeio, 
s K T, and E' is \RT. The fact, howevci, that y falls, with using 
Lcmpciatiuc shows that the total encigy docs not picscrvc a con- 
tain 1 idtio lo Lhe eneigy of tianslation, and hence that theic 
is not equipaitition of the eneigy among the possible modes of 
motion 
In any gas we may wnte 

E=- E' + E" + E'". 

The cneigy of tianslation E' vanes as T, being equal to }RT. The 
eneigy of jotation E" bcais, in any given type of molecule, a 
n onslaut latio to E', and Iheicforc also vanes as T If the eneigy 
ol \ibiation E'" also boic a conslant ratio to E', Lhe \\hole encigy 
vsould Aaiy as 7\ \\luch is inconsistent with Lhe cxpeinnental 
(.Mills staled in Chap VI 

224 Energy of Vibration The tcim E'" includes not onl\ 
.IK.'] yy due lo \ ihuilions ol the c onslilucnl atoms iclativcly lo one 
molhci \\illun Ihc 1 niokc'iik (//") but encig\ due lo \ibialions 
[IIION <. menls ol c Ice lions) willun I he consliluciit atoms LhcmscK es 
Jl a '") It is known lli.il l'] a '" is <i \eiy small i),uL of the whole 
iicioy, even at lempeuiliiics as high as 2000 C 1 The \ibialions 
hat m.ike up E tl '" have much highci fiequcncics than those that 
nake up A 1 ,,/" [I is lo \ ibiiitions willun Lhe consliluc.nl atoms that 
)iie alliibuLes Lhe bnghl lines which make up Lhe visible spcctium 
>l nn incandescent gas, and Lhe coiicsponchng daik lines due lo 
ibsoiplion in Lhe visible spec! i urn of light tiansmiltcd Ihiough 
i cold gas The longer-pcnod vibrations that make up E m '" cmil 
>r absorb rays which he in Lhe infra-red legion, beyond the range 
>L the visible spectrum. It is Ihesc longei -period vibiaLions thai 
'onstiLutc Lhe mam part of Ihe vibiatory cncrg} r when a gas is 
.trongly heated, as in a llamc or an explosion, and give rise to 
nost of its radiant energy. 

Fiom the theory that has been outlined above, of the consti- 



362 THERMODYNAMICS [APP. 

tut ion of a diatomic molecule, we should expect il lo li<i\ r c one 
well-maiked period of fiee \ ibiation, and thciclbic Lo show a stiong 
emission band when heated, 01 when excited by elect nc dischaigc 
in a vacuum tube, and also a coiiespondmg sliong absoipLion 
band when cold. A good example is furnished by caibonic oxide 
(CO), whose infia-ied spec ti um is found to consist almost entirely 
of one chaiacteiistic band, the wa~\ c-length of which is nbout 
4 7 /JL ^\hen the gas is emitting uidiant cncig^, and I O^u, when 
the gas isabsoibmg it ! The fact that these wave-lengths aie so 
neaily the same is, evidence that what may be called the stiffness 
of the quasi-elastic link bctueen Lhe atoms, due to chemical 
affinity, suffeis little change when the gas passes liom the cold Lo 
the ladiant state 

Again, in the mfia-ied spectrum of the tnatomic gas C0 2 we 
should expect to find three piomment bands coiicspondmg lo the 
thiee modes of vibiation that can be scl up within a CO^ mole- 
cule by iclatne movements of the caibon and oxygen atoms) 
This is in agieement with \\l\at is obsu\cd Then- au, both in 
absoiption and emission, thiee distinct mfia-icd bands, namely 
a strong band whose wave-length is about I Ifj,, a weak band 
with a wa\ e-Iength of 2-7^, and anothei with a much longei \va\ e- 
length, between ll^and 15/j, (Ait 173) This long-pet iod \ibia- 
tion accounts foi the fact that c\ en at oidmaiy tempeialiius I he 
specific heat of C0 2 exceeds the value it Mould ba\ c 1 il theie weie 
no vibiatoiy cneigj, making y distinctly less than 1 ,W,*> Koi the 
pimeiple holds that \ibiations of long peiiod icqime no mou llian 
a compaiatn cly low tempeiatuic to excite them into taking iij) 
someconsideiable shaie of the cneig}', so that they then conliilmlc 
substantially to the specific heat, wheieas those ot shot I pcuocl do 
not begin to take up an appreciable sluue until the g.is is sliongly 
heated 

225 Planck's Formula This pimciple Imds cvpussmn in a 
lormula de^ iscd by Max Planck to connect the oiurgv of any 
paiticnlar hequency of vibiation with Lhe hcqnency and wilh 
the tempciatme, when a .slate of cquilibiium has been icached 

* See W \V (Joblent^, Invrslirjutions of Iiifxi ml tfpu,lnt I'uhln ulioiiti of tho 
Carnegie Institute, Washington, No 30 1005, No dS, ]<)()(>, No !)7, 100S 

f N Bjeiuim, Vo> handlii ngc.n da dculsJicn Plu/t, GcsfllstJinJI, 1914, p 737, ilis 
ciib^e^ tlie lijpothetiral ronfiffuiation of a COj molecule \\hich would vibuito with 
penocls con expanding to the three obseived vAave lengths, whuli ho takos as 2 Ip, 
i 3fj., and 14 "/JL 



[i] MOLECULAR THEORY OF GASES 363 

Accoidmg to Planck's theoiy the \ibiatory cneigy, per mol, 
coiiespondmg to any particulai frequency v is' 1 

!,_ N ^ 



e RT 1 

Hcic N l is the number of molecules per mol, namely G 10 x 10 23 , 
anel h is a constant, kno\\n as Planck's constant, which is the same 
for all gases and is appioximatel} equal to C 55 x 10~ 27 e^s R, as 
usual, is the gas-constant, whose value pei mol is 1 985theimal units 
01 8,3 1 x 10 eigs, and e is the base of the Napieiian loganthms, 
"2 71828 The frequency v is equal to c/A, wheie c is the \elocity of 
light, or ,'3 x 1() 10 cms pci second, and A is the wave-length in cms. 
In a gas whose molecules aie capable of moie than one mode of 
vibration the whole vibiahonal eneigy E'" would be the sum ol as 
many lei ins, in Ihc above foim, as aie lequncd to expiess the 
\auons modes Thus in caibomc acid, for example, theic would 
be thiee tcims lot lieeiuencics e>f yibiation corrcspe)ndmg to the 
lluce obsened \\a\ e-leni>ths 

Vl any one dequeues v, lei the quantity NJiv/RT be upiesented 
bv ,(.' The n -j jff 



<ui(l Planck's loimula bcconu's 



\vheie ' is a laeloi the \ aluc. ol \vhieli depends on Ijoth v and T 
& { 1 

lot any gi\en^ it lends to an uppei limit ol I when T is indefinitely 
mcieasedand lo a louei limil of /no when T is indtlimlcly leducerl 
Hence, if we accept I he loimula as valid, il follows lhat when I he 
molecule's of a gas aie dee lo vibialc in any one mode, the gas will 
lake up, m respect of I hat Uecclom, a quantity of eneigy \\bich 
appioMches the limit RT when the gas is sliongly he'alexl. This will 
l)e- I me also of .my olhei mode office vibiahon which I he molecules 
possess When Ihe gas is healed lo any given lempei aline the 
Jiaclion of RT which is laken up will in ocneral be difleienl foi 
different modes of vibialion, for il depends on the frequencv, being 
smallci when the ficcjuency is high This, aceoiding lo the Ihcoiy, 
is why the high-ficquency modes oJ vilnation Avhich aie ic\ca,le.d 
by the visible spec ti um do not contribute substantially to the 

* Foi a discussion of the tlicorotical basis of Planck's fonnula, see Joana' 
Dynamical Tkconj of Gases, Cliap XVIII 



364, THERMODYNAMICS [AIM' 

whole eneigy of a gas, e\en at tcmpciaturcs such as ;nv teaelu (I m 
an oidmaiy flame 01 in a gas-engine explosion, ami why, in llu- 
leckonmg of eneigv and of specific heal il is only vibialums i>( 
mfia-ied frequency that need be taken into account. Koi (he same 
reason a gas whose molecules have one 01 moic loiiif-pmod lypes 
of vibration may, at oidmarv tcmpeiatuics, hold u considerable 
quantity of eneigy in the vibiatoiy foim, and have i spceilic lual 
markedly gieater than the ideal (vibialionless) vnlnr. 

The amount by which any one mode of Mbiahoii will augment 
the specific heat is found by diffcicnlialmg (-\\ilh ies|K-cl Io T) I he 
expiession for the extra internal cncigy lh.il is due- Io Mini mode 

We may wiite it ,77? f < ,,j 

ITS \ _ UiJl _ c _!! if 

A "~~ l 




12 14 1C, 



Heie is a factor, depending on I he wave-length and I he 

tempeiatuie, which langcs fiom zeio Io unily as llic <|iianlil\ 
l/x is inci eased fiom zeio to infinity Fig, 90 exhibits (he maiiiK i 
in which this factoi incieases icUtively to I/a 1 . It shows thai llu ic 
is a veiy lapid use in the faclor, and Ihcicfore in I he speeilie heal, 
aftei l/x has i cached a value of aboul 1, bul up Io Hint pom! the 
effect of vibiation on the specific heat is quite insignificant. At 
C. the value of A \\hich coiicsponds to I/,/ 1 = 01 is OOOO.Ili, 
hence it is only those modes of vibiation whose wave-lengths are 
greater than say 5/z that sensibly affect Lhc specific lieal of a gas 
at normal tempeiatuie. 
As an example, take the diatomic gas CO with its eharaoleusf ic 



] MOLECULAR THEORY OF GASES 365 

bullion foi which A is about I- 7/i or 00017 cm. Foi that wave- 
nglh the value of I/a;, at C, is 09, and at 2000 C it is 74 
ho factor c' t r 2 /(e l - I) 2 is theiefoie nisignificanLly small at C., 
it becomes about SO at 2000 C Hence the calculated specific 
\il K v , \vhich is }R at C , uses, as a consequence of this 
bialional cncigy, to ( r \ + 86) R at 2000 C , and the coircspond- 
i value of y falls from 1 l< to baiely 1-3. 

Again, take the tuaLomic gas C0 2 , one of whose chaiacleiistic 
ibialions has a, wave-length of nearly 15/x So slow a vibiation 
)ntiibulcs snbslantially to the specific heal even when Ihc gas is 
)ld At C a wave-length of 15/j, makes I/a 1 = 021 and 

e^ l2 /(e fl - I) 2 = 028. 

encc a single mode of ^ ibiation with that ficqnency should bung 
ic specific heat ol Ihc cold gas up to about (3 + 28) R, and 
cluce y horn 1 333 to 1 305 When the gas is stiongly heated, 
count has lo be taken of tluee modes of \ibiation whose wavc- 
nglhs aie long enough to be impoitant In itspeet of the thiec 
)gclhei, K obviously lends, at \ ei^ high lempeiatuies, to in- 
case to\vauK a linul of G/i', and y lo (all to ^, a,pai t liom an\ - 
ling thai olliei \ ibialions may conliibuie, and apail horn effects 
I dissociation 

Though I he ideas undeilvmg Planck's Lluoiv aic open to 
spulc, lh< ic c.in be hltk doubt that a, cui\ r ( moie <n less like 
lal ol li<> ')') docs u[)Kscnl the way in which molecular 
biiition ol a gi\ c'n l\pc eouli ibutes lo the specific IK at At 
si, when llu gas is being luale'el liom a cold stale, the conlii- 
ulion is piache.dh ml, MUM theic is a shai]) use, and finallv 
i as\ m[)lol ic appioach lowiiuls a limit The UmpciciliiK at 
Inch Hie sliaip use 1 begins dc [>e nels upon Ihc ficqnency ol liee 
ibi.ihon, being lugliei when Ihc dequeues is high 
The I'ael thai in pol\ atomic gases geneiallv the specific heals, al 
iinal li'mpc lahnc, aicgiculci lhaiitlu ielecLl(v ibuitionless) \ allies, 
id y is nolably less lhan 1 3.33, is to be asciibcel lo Ihcn [)ossessmg 
ug-peiioel modes of vibiation which aie icsponsive to low- 
mpeiatmc encouulers A complex polyatomic molecule may 
ive many such modes, cnch producing a substantial augmen la- 
mi of Ihc specific heal 

Smuiailv Ihc chaiactciistic mode of vibiation m a diatomic gas 
ay be so slow as lo affect the specific heal at noimal 01 com- 
nativcly low tcmperatuic, making K v gicatci than r \ll, and K v 



366 THERMODYNAMICS | -vri' TI 



gieater than ^R, and y less than 1 i< This is nolably I he case 
the vapoins of the halogen elements C1 2 , Bi 2 , 1 2 These cleinenls 
have high atomic weight, and it would ,sccm lhal in each of them 
the pan of heavy atoms in the molecule, pcihaps lather loosely 
held together, have a slow type of vibialion, which explains the 
obseived high specific heats and low value of the latio y When 
a hydrogen atom is substituted for one of the pan, Ihis chaiaclei- 
istic disappear^, foi the gases HC1, IILh and III, when cold, <ue 
found to have specific heats that appioximate to tlic normal values, 
with the latio 1 4. 

226. Effect of Extreme Cold on the Diatomic Molecules 
of Hydrogen. It has been found that Avhcn hydiogen is cooled 
to about -200 C. its specific heat falls piogiessncly to n, value 
not much gieatei than that foi a monatomic gas, and y uses 
to a value not much short of that foi a monatomic gas (1 667). This 
lemaikable lesnlt, fiist obseu ed m mcasuicmcnts of K a , has been 
confhmed by independent measuiements of K and of y ' . It 
appeals theiefoie that under extiemc cold the hydiogen molecule 
tends to assume a diffeient stiuctme, becoming in effect quasi- 
monatomic, piesumabh by the coalescence of the tuo atoms which, 
at oidmaiy tcmpeiatuies, aie held apait The pan ol atoms 
appaienth behave as if the foiees \\luch usii<ill\ liolcl UK ma pa it 
A\hat we called then tender in Ait 223 cease to be ellecinc m 
pieA entmg the massive nuclei fiom coming togethei, to loim 
what is vntually a single-atom molecule ol double mas--, It may bo 
conjectmed that this happens Avheu tlic lotational speed ol I lie 
diatomic molecule tails belo\\ a ccitam limil, ami llial tlu moltoule 
thenietams the coalesced state until its cons ti hie uL atoms aic loieeel 
apait by a snlhcicntlv \'iolent cncountci \\iule it umaiiis m I he 
quasi-monatomic state it takes up cneig^ ol liansl.ition onl\, and 
when a laige piopoition of the molecules aic m lhal slale I lie 
gas beh^ es appioximately as a monatomic gas in icspect of ils 
specific heatb So fai as is known this action is peculiai lo hydiogen, 
it does not occui in oxygen, mtiogcn, 01 caibomc oxide 

* Eucken, tiitzunysbenchte d L Pieuss Akad , Boilin Fcb 1012, School and 
Heuse, do, Jan 1913, also Ann d Phyt>il, Vol 10, p 473, 1<)1J, M SJuelda, 
Phijs Review, Nov 1917 



APPENDIX III 

'FABLES OF THE PROPERTIES OF STEAM 

ihlc A Piopcilies of Sal mated Steam, in i elation to the 

Tcmpeialmc 

A 1 . Piopeitics of Walei a I Sat i nation Picssuic. 
1? Piopulies ol Salmatul Slcam, in iclalion to the 

Piessme 

Volume ol Sleam in am Div Slate 
Tolal Heat ol Steam in any Div Stale 
K Knhopv ol Slcam in any Diy Stale 
I' 1 Spctilie Ile.il, al conslant picssiue, ol Skain in any 

I)i\ Stale 

Tin sc Table -1 an based on Callciulai\ i'oiinnlas 3 and will scnc 

illii^lial( Ins mi I hods Tin (ionics aie, lot the most pait, taken 

run Thf Ctil/cncldi Sttmn Tahiti published b\ Ed\\anl Ainold, 

)l r ), \vlu< h will be loimd lo ni\e much moie complete paiticulaii 



368 



THERMODYNAMICS 



[APP, 



TABLE A Piopotiet, oj Sal in tiled 



Temp 
Cent 


Pi 03311 10, 

pounds pei 
sq inch 


Volume, 
cub ft 
pei Ib 
V, 


Tol.il 
Heat L 
Ib calo- 
nos pei Ib 


hltlOp\ , 
pei II) 


LfiU.nl 
Jloat, 
II) culn- 
IICN poi II) 
L 


Inlonul 

11) (illo- 

uis pi i Hi 





00892 


3275 9 


594 27 


I 17602 


59127 


561 21 


10 


1788 


1693S 


59901 1 


> 11 OH) 


589 ? 


507 85 


20 


03399 


922 19 


603 72 : 


20l>221 


58 ) 78 


571 48 


30 


00162 


525 81 


608 40 


> 01247 


578 19 


575 07 


40 


10703 


31245 


01304 


%088 


573 15 


57804 


50 


1 7888 


19272 


0170') 


92490 


567 75 


5S2 17 


60 


28873 


12291 


022 10 


88021 


562 29 


585 00 


70 


4 5156 


SO 804 


020 00 


85039 


556 72 


589 07 


80 


6 8627 


54 596 


030 95 


81712 


551 05 


5<>2 4 1 


90 


10 101 


37 815 


035 19 


78019 


5 15 25 


595 07 


100 


14089 


26 789 


039 30 


75732 


539 SO 


598 S3 


110 


20777 


19370 


64/20 


73027 


533 17 


001 80 


120 


28 SOS 


14271 


647 07 


70-185 


520 85 


001 78 


130 


39213 


10 090 


(.50 72 


08092 


520 i2 


007 58 


140 


52 482 


8 1431 


054 19 


65S3I 


5 1 J 57 


01023 


150 


69 150 


6 2895 


657 47 


0368'! 


500 50 


01273 


160 


89800 


4 9232 


660 55 


01057 


499 29 


01508 


170 


11506 


3 9015 


663 44 


5972 1 


191 75 


01727 


180 


14559 


3 1275 


066 14 1 


57884 


183 <3 


01930 


190 


18208 


2 5339 


608 05 


50128 


47582 


02 J 19 


200 


225 24 


2 0738 


670 96 1 


51453 


407 1 1 


022 91 


210 


275 78 


17134 


073 09 


52851 


458 09 


02^1 18 


220 


334 38 


1 4285 


675 00 J 


51320 


119 09 


(.25 03 


230 


40189 


1 2007, 


676 87 


19868 


J40 38 


027 23 


240 


47874 


1 0178 


678 55 1 


I 4S480 


43081 


028 43 


250 


56563 


8695 


680 12 J 


L 4710 1 


(20 90 


029 53 



TABLES OF THE PROPERTIES OF STEAM 



3G9 



Piopeiliei, oj IVaio at Salutation 



nip 
ont 


PtC-HIILO, 

pounds poi 


Volume, 
cub ft 
poi 11) 


Total Heat, 
Ib -calouea 
pei Ib 


Enliopy, 
pei Ib 


Function G, 
Ib caloiica 
pei Ib 




J'. =1>* 


v* 


/, 





<*u=-G* 





00892 


01002 











10 


1788 


001003 


998 


03585 


0181 


20 


03390 


001005 


1091 


07040 


0714 


30 


00102 


001009 


2991 


10393 


J 58 


40 


10703 


001014 


39 89 


13031 


278 


50 


1 7888 


001021 


4088 


10770 


4 30 


60 


2 8873 


0011)29 


59 S7 


19815 


13 


70 


45150 


OOlolS 


0088 


22774 


826 


80 


8027 


001048 


70 90 


25052 


10 OS 


90 


10 101 


01059 


80 94 


02S151 


1338 


00 


I I 089 


001071 


100 00 


3118ft 


10 30 


10 


20 777 


001081 


11000 


33853 


10 00 


20 


28 80S 


001008 


120 22 


30400 


21 10 


30 


39213 


001713 


13040 


030011 


20 81) 


40 


52 482 


001729 


140 02 


041511 


30 SO 


50 


00 150 


001740 


1500] 


(301)} 


3 r > 10 


60 


89 800 


001705 


10! 20 


10 {7$ 


30 5S 


70 


1 15 00 


001785 


17! 00 


U487H 


44 2<) 


80 


145 59 


001807 


18221 


051078 


19 22 


90 


182 OS 


001831 


192 83 


053381 


5 I 38 


00 


225 24 


001850 


203 55 


55054 


50 75 


10 


275 78 


001885 


214 40 


57004 


O r > 13 


20 


3 J t 38 


001014 


225 37 


00 1 28 


71 12 


30 


401 80 


001940 


230 49 


023 12 


77 11 


40 


478 74 


001980 


21774 


004517 


8330 


50 


505 03 


002010 


259 10 


OOOS7 


89 OS 



THERMODYNAMICS 



[APP 



TABLE B 



of Katiiifitid 



1 

Piessuic, 
pounds 


Temp 


Volume, 
cub ft 


1 Total 
Heat, 


Knliops 


L.Lll 111 

limit, 


' h'lmUun 
(1, 


pei &q 


Cent 


1">AI ])"> 


11) calnuos 


pei Ib 


III C ll()l 11 S 


Hi t,il<> 


inch 


t 


IJlsL Lt~r 

V 


pci Ib 


^ 


poi 11) 


1 11 S ] 1C 1 [ 


P 




' & 


I* 




L 


C/,, _ (,' w 


01 


1 59 


2940 


595 03 


2 J662 


593 4 1 


005 


0-2 


J 1 09 


1524 


599 81 


2 1068 


5S8 15 


02-16 


0-3 


1799 


103S 


602 77 


2 0727 


58 t 83 


058 


0-4 


2206 


7907 


604 97 


20182 


582 3b 


091 


0'5 


2641 


0505 


606 73 


2 0299 


5SO 10 


1 23 


1 


3874 


333 J 


61246 


1 '172-1 


573 8 ! 


261 


2 


52 27 


1735 


618 67 


1 ( H5 ( ) 


56(> 52 


1 69 


3 


6083 


1186 


62253 


I 8S } ! 


561 Si 


(. 30 


4 


0723 


9054 


625 38 


I 81)00 


558 28 


701 


5 


7238 


7344 


627 1 


I SI 22 


555 !S 


881 


6 


7672 


619] 


b2<) 52 


] 8277 


552 92 


<) SI) 


7 


8049 


53 59 


631 15 


I 8 11I> 


55071. 


1081 


8 


8384 


4730 


632 57 


1 8019 


5 18 82 


11 69 


9 


8684 


4230 


033 85 


1 7951) 


547 0< 


12 50 


10 


8958 


3839 


635 01 


1 7S71 


515 50 


13 2(. 


12 


94 14 


3237 


037 02 


1 7731 


542 61 


I 1 07 


14 


986(3 


2802 


638 77 


I 7611 


510 12 


15 'II 


16 


102 11 


2473 


640 26 


1 7506 


5 57 S ! 


17 12 


18 


103 79 


22 16 


641 60 


I 7111 


5 {5 7 r > 


IS 20 


20 


10887 


2008 


6L282 


1 73 !3 


533 S7 


l')22 


22 


11171 


1837 


64392 


1 72 5S 


532 0') 


20 Ih 


24 


11434 


16 93 


GU 93 


1 7 1 89 


550 1 1 


21 09 


26 


11080 


1571 


15 85 


1 712b 


528 SS 


21 ( > r > 


28 


11911 


1460 


04071 


] 7069 


5J7 J2 


22 7S 


30 


12128 


1374 


647 5 1 


17016 


526 02 


23 51. 


32 


123 35 


1294 


618 JO 


1 6<I66 


52 1 67 


2 1 33 


34 


12531 


12 22 


649 02 


] 6919 


523 10 


25 07 


36 


127 17 


1159 


649 69 


1 687 J 


522 17 


25 77 


38 


12896 


1102 


65031 


1 6831 


521 00 


21) 15 


40 


13007 


1050 


650 95 


1 6792 


51987 


27 12 


42 


13231 


1003 


651 53 


1 6751- 


518 77 


27 76 


44 


133 89 


9003 


652 OS 


1 6719 


51771 


28 40 


46 


13541 


9212 


652 61 


1 0685 


516 68 


29 00 


48 


13688 


8853 


053 12 


16651 


515 69 


29 59 



TI 



TABLES OF THE PROPERTIES OF STEAM 371 



(coHltnut'il) 



of tiatiuatcd Stcant 



pounds 


Te nip 


Volume 1 , 
i ii It li 


Heat, 


l <1 ntiop\ , 


Latonl 
HcMl, 


KlIIKllOll 


put Hq 


Cent 


L> 1 1 f J L (j 
11 


11) lalo 


pin Ib 


11) i-nlcim* 


i Ib Gciloucs 


moll 


t 


poi II) 


MLS pen Ib 




p(?T 11) 


poi Ib 


l> 




^ 


I* 




J 


('i = flw 


50 


138 30 


S 520 


053 00 


1 0020 


514 71 


3010 


60 


1 14 79 


7 JSt 


055 77 


1 0479 


51022 


3285 


70 


15046 


(5218 


057 (31 


1 0359 


500 23 


3530 


80 


155 52 


5 187 


059 20 


1 0250 


502 59 


3754 


90 


10009 


4 913 


000 59 


1 0105 


49924 


39 02 


100 


1(51-28 


1 451 


001 82 


1 0082 


496 11 


41 58 


110 


108 15 


4 070 


002 93 


1 0007 


493 18 


4340 


120 


171 75 


3 75 1 


0'03 92 


1 5938 


49040 


45 13 


130 


175 13 


3 179 


0(54 S3 


1 5875 


487 70 


4078 


140 


17831 


'3 245 


005 09 


1 5818 


1-8527 


4837 


150 


181 51 


! 041 


oor. 49 


1 571)5 


182 90 


4989 


160 


IS! 10 


2 S02 


007 22 


1 5715 


48001 


51 34 


170 


ISO 88 


1 70 { 


007 90 


1 5000 


178 40 


52 75 


180 


1S9 IS 


2 502 


(.OS 53 


I 5020 


170 20 


54 10 


190 


191 97 


2 135 


(.09 J 3 


1 5577 


17 1 20 


55 12 


200 


i9rr> 


2 '!20 


009 <><) 


1 5538 


472 21 


50 09 


210 


I ( M> hi) 


2210 


07020 


1 5502 


470 20 


57 94 


220 


I9SS7 


2 120 


070 70 


1 540". 


108 38 


59 13 


230 


20! 02 


2 54 


1)71 19 


1 5129 


10055 


0031 


240 


20'3 <)<> 


1 95 1- 


071 04 


1 53% 


40170 


01 45 


250 


205 10 


I SSO 


072 07 


1 5302 


10300 


02 5S 


260 


207 01 


1 SI 1 


072 IS 


I r . \ \1 


401 30 


03 00 


270 


21 IS 9 { 


1 7 IS 


072 SS 


\ 5 JO i 


459 05 


04 72 


280 


210 77 


I OS') 


073 25 


1 527 t 


158 02 


(.5 77 


290 


21257 


1 Oil 


07'} 01 


1 52 10 


150 11 


00 79 


300 


2 1 \. 32 


I 5S5 


073 90 


I 5219 


151 St 


07 SO 


350 


222 15 


1 308 


075 52 


I 500(5 


11741 


7257 ! 


400 


229 75 


I 200 


070 8 ( 


1 49!)l 


410(53 


70 9(5 


450 


23(5 12 


I 07') 


(.77 97 


1 1897 


43 1 28 


810(5 


500 


2 12 57 


0977 


(578 97 


1 J-8H 


428 31 


SI 92 



242 



372 



THERMODYNAMICS 



vrr. 



TABLE C Volume, in cubic jeet pet U> 





Pieasiue in pounds poi sq inch 


Temp 












Cent 














20 


40 


60 80 


100 


120 140 


400 


35988 


17973 


11967 89618 


7 1632 


60009 5 10-13 


350 


33 295 


16617 


11058 82785 


66107 


5 1989 | 1 701-8 


300 


30 594 


15 254 


10 141 7 5848 


6 0509 


5 0284 1 2980 


290 


30 052 


14981 


9 9569 7 4449 


5 9 {78 


JfJUO 12153 


280 


29 510 


14706 


97718 73045 


58241 


48372 4 1323 


270 


28 967 


14431 


9586? 71630 


57100 


47109 10-187 


260 


28 425 


14 156 


94002 70221 


5 5953 


4 (>! II 


3 9040 


250 


27881 


13 880 


9 2134 6 8799 


5 4708 


45105 .58798 


240 


27 337 


13 603 


90260 6 7370 


530^7 


4 1 KSI 


3 7<H2 


230 


26791 


13 326 


8 837i > 


6 'V)33 


5 2-1 Ii7 


4 5190 


3 7078 


220 


26 240 


13 048 


8 6483 


6 4480 


5 1289 


-1 2190 


'! 0205 


210 


25 699 


12768 


8 4582 


6303L 


50101 


1 MM 


3 5324 


200 


25 150 


12488 


8 266S 


6 15()|. 


t8')OI 


1 0159 


3 mo 


190 


24601 


12206 


80743 


( 0085 


I 7690 


!'II27 


! 3524 


180 


24 050 


1 1 923 


7 8805 5 85 l >2 


4 61 65 


3SJ80 


{ 2000 


170 


23 497 


11 638 


7 6350 5 7083 


4 5J2 t 


37!I7 


3 1070 


160 


22944 


11352 


74878 55558 

1 


4 3%0 


i 02 {8 


30718 


150 


22388 


11063 


1 1 
72886 | 54012 


-1 2687 


3 r> 1 !8 


29715 


140 


21 829 


10 773 


70871 


52413 


1 1380 


3 IU|(, 


2 875 1 


130 


21 268 


10479 


6 8832 


50850 


J-OOOL 


32809 27731 


120 


20 705 


10 183 


66762 


4 9227 


3 8706 


3 1791 2 0081 


110 


20 138 


9SSJO 


64661 


4 7572 


37J1S 


30182 25599 


100 


19 567 


95809 


6 2522 


4 5878 
i 


3 5892 


29235 2 1180 



] TABLES OF THE PROPERTIES OF STEAM 373 



of Si cam ui any Dnj State. 



111 pounds poi si| inc.li 



snip 
















'out 


160 


180 


200 


250 


300 


350 


400 


100 
350 


(- 4()() f ) 
1 1001 


.'{ flfiOf) 
> (>t59 


3 5I>01 

.i 27r>j 


2 8,wr> 

2(>08l 


2 JflOl 
2 K.IJ'j 


20 189 

1 SJ58 


I 7586 
1 b075 


300 
290 
280 
270 
260 


'{7,101 
, (>77I 
, ()()}(> 

r>2' n 
l r ), r )() 


.5 :5240 
j 258 r ) 

,j l')2l 

:: 1 2 is 

>05S7 


2'KSil 
2 02,ir 
2 S(. !4 
28027 
2 7 M(> 


2 ]()<)(> 
2 J20I, 
2 2712 
2 2 '1 2 
2 170') 


XiOG 
<)IS7 
S7(>5 
S i,'57 
7,V)4 


1 (>(iS,{ 
1 (>,]17 

i r ( )4r> 

1 r ). r )(i8 
1 JllSI) 


1 44!)2 
1 4K.4 
1 JS'50 
1 .'1401 
I ^18 


250 
240 
230 


,57'i7 
, 5057 
, 2200 


2 0008 
2 '22 
2 h.-20 


2 (7 l l7 
'2 (.171 

2r>fi!i 


2 I I'll) 
201)77 
201 t'l 


7I(J 
70 l r > 
() r )") ( ) 


1 4701) 
1 HJ!I<) 
1 Wi 


1 2701. 
1 2M7 
1 2071 


220 


, 1 I'U 


2 7S2<> 


24VH 


1 i!4 


1 I)0'I4 


i ir.so 


1 100~) 


210 


j 070(i 


2 7 1 1 -1 


2 1211 


1 'HlthS ' 


1 r .(.20 


I iir>8 


1 1 !IO 


200 


2 0008 


2 (){'() 


2 J. r j7( 


1 S r ,l 1 " 


1 ill f. 


1 272 { 


1 001 I 


190 


2 ( )0')7 


2 5<>5 r > 


2 2 f H)it 


1 7'MI 


1 H!7 


1 2271) 


1 OHOl 


180 


2 h27l 


2 lOOli 


) j > |o 


1 7 :i>0 


1 II2(. 


1 ISK) 


1 008! 


170 


27(51 


2 mo 


2 noi 


1 <>7f>() 


1 r>')S 


i i t,:<) 


Olil r > 


160 


_> I)T7S 


2 i.ns 


2 07S2 


1 dl l r > 


1 I0 r )l 


1 OMI> 


00100 


150 


2 r 70l 


2 2'~>5 r ) 


2 00!0 


I r, r (is 


1 218') 


1 O.H2 


08714 


140 
130 
120 
110 


2 IS02 

2 W7S 
22021 
2 1057 


2 17.51 
20SSI 
2 0001 

I DOS') 


1 <)27 5 
1 SIS! 
I 7((. ! 
1 <>SIO 


1 IS'll 
1 II (.7 

I nr>i 

1 270S 


1 l')02 
1 I2'IO 
1 01) l<) 
( )')7r> 


0'I7<)() 

o <)2:ir> 
os()ir> 

S022 


S2 1 7 
071)01 
071 M 
<) r ) r >7 


100 


2001:5 


1 SI 10 


1 5 ( )2() 


1 I')2(> 


920 ! 


07l()I 


50,54 

1 



374 



THERMODYNAMICS 



[AIM 1 



TABLE D Total Heat /, in II) -colon es pet Ib 



Picsauic in pounds poi s<| nu'li 



Temp 
Cent 










20 

78470 


40 60 80 100 


120 

7S2 23 


110 

781 71 


400 


78420 78371 78322 78273 


350 760 G8 76004 75939 75874 75810 75715 


7f><> SI 


300 736 61 735 74 734 88 73 t 01 7 ! ! 15 7 52 2S 75112 


290 73178 73086 72994 72902 72810 727 IS 721. Jd 


280 72695 72597 72499 72402 72301 722 0<> 721 OS 


270 72211 72107 72003 71899 71795 710 01 7IOS7 


260 


71727 71016 71506 71305 71281 71173 7I<>(>2 


250 


71243 71124 710 Od 


708 S'7 707 (.0 


706 51 


71 H !2 


240 


70757 70631 


705 04 


703 7S 702 ".1 


701 21 


(,(,() 4)1) 


230 


70271 


701 36 


700 01 


608 ()(. 


d07 !0 


(.01 <I5 


d<> / <)() 


220 


69785 


606 40 


691 05 


(.0 3 50 


(.02 05 


600 (.0 


dSM Id 


210 


69297 


691 42 


680 86 


(.8S !l 


dSd7d 


(.85 20 


dS ! (>-> 


200 


688 08 686 42 


684 75 


(.83 08 


(.Si II 


(>70 71 


(.7S OS 


190 


683 19 


681 30 


679 60 


(.77 SI 


<>7dOI 


(.71 22 


(.72 1 ! 


180 


678 28 


676 35 


(.7141 


d72 48 


07055 


66S 62 


()(>(. (.0 


170 


673 35 


671 27 


669 10 


6(.7 10 dd r . 02 


ddJ ')! 


~ddO M" 


160 


66841 666 Id 


(.6391 


661 66 


650 II 


(.57 Id 


(.5 1 Ml 


150 
140 
130 
120 
110 
100 


663 46 
65848 
65348 
64846 
64340 
6,38 31 


661 02 
65584 
~ 650 61 
645 34 
64000 
63459 


658 59 


(>5d 15 
650 56 
6-1 I 88 
6'39 00 
63'3 10 
627 15 


61'! 71 
61702 
61201 
6 !5 07 
620 79 
623 i,{ 


dll 2S 
(.I52S 
d.i!) 1 1 
(. !2 S5 
626 3S 
(.1971 


(.IS SI 
6!<> "21 

dir> ')' 


653 20 
6 17 75 
6-1221 
636 50 
630 87 



TABLES OF THE PROPERTIES OF STEAM 375 



aj fitiani in a in/ Dnj Stale 



J'ICV-HIIO m pounds pei srj 


inch 


'cm]) 


















'cnl 




















160 


180 


200 


250 


300 


350 


400 


400 


78 1 2,1 


780 71. 


780 2<> 


77!) 0,} 


111 


80 


77(i 57 


77.1 U.J 


350 


7fid Id 


7 15 ill 


71487 


7.13 25 


7,11 


di 


750 02 


748 10 


300 


7 50 ill 


720 dO 


728 82 


72d dd 


724 


.10 


722 .5 t 


72(1 17 


290 


721 ,14 


721 I,} 


72. if. 1 


72 1 2 1 


7 IS 


0] 


7lddl 


7lf Jl 


280 


720 1 1 


710 1 1 


7IS 1,1 


7 1 ,1 7 1 


71 i 


2d 


71082 


70S 57 


270 


7 1 1 S ! 


71 ! 70 


71271 


710 14 


707 


11 


704 01 


702 >4 


260 


700 11 


70S 10 


707 20 


701 12 


701 


7.1 


dOh OS 


(.0(. 21 


250 


7<H U 


702 01 


701 77 


I.OSSI 


(.0.1 


hi 


d02 SO 


dSO j 


240 


(.OS 72 


(.07 Id 


dOd 10 


dO { ll.{ I.S'O S7 


dS'd 70 


dS i 14 


230 


(.0 i 2t 


dOl SO 


dOM,14 


dS7 Id 


(..S i 


T7 


dS'O 10 


(.77 01 


220 


<S7 71 


dSd 2l> 


dM SI 


LSI l<) 


(.77 


r.d | 


1.7 JOJ 


d70 ,!2" 


210 


dS2 10 


dSO .U 


d7s 0<) 


1)7". 10 


(.71 


i > 


dd7 !l 


dd! 11 


200 
190 


(>7d 1 1 
d70 (>.] 


(.71 71 
ddb SI 


-H'-!f- 


ddS OO 
d(>2 Id 


ddl 

dlS 


OS 


ddO f>d 


did .50 
MM II 


180 


(id 1 7d 


dd2 SI 




did 07 


dll 


21 


Old II 


dll IS 


170 


dlS 77 


did (.0 


d,l 1 dl) 


1.10 10 


dl 1 


10 


d {S OS 


di! 77 


160 


(,12 dd 


(.10 II 


dlS Id 


(.12 l'{ 


d id 


01 


(ill 2S 


(.2;! dd 


150 


did II 


(>H 07 


dll 1{ 


d i! 14 


(>20 


,!,! 


I.2.J 2<> 


(.17 17 


140 


dlOOO 


(.17 .!(. 


d.il 72 


(>2S 12 


(>2I 


12 


(.1 1 02 


dOS ,!2 


130 


(. i ! II 


(.,!() 11 


(.27 1.7 


(i20 10 


(.I.! 


n 5 


dOd Id 


508 00 


120 


(i2d dl 


(.2.! IS 


Ii20 id 


(.12 Id 


(.01 


71 


lOd 01 


.ISO 1.1 


110 


dlO .17 


did 17 


d!2 77 


(.01 2d 


101 


71 


1H7 "2 1 


57.S 7.} 


100 


(.1227 


dOS 11 


d(M S { 


,10.1 ,12 


5Sd 


22 


17d 02 


,ld7 (.2 



37(5 



THERMODYNAMICS 



TABLE E. Enhopy </> of Steam 



in pounds poi sq inch 



Temp 












Cent 














20 


40 


60 


80 


100 120 140 


400 


20100 


1 9331 


1 SS78 


1 8556 


1 S304 


1 8097 , 1 7921 


350 


1 9729 


18958 


1 8503 


18178 


1 7924 


1 77H J 7530 


300 


1 032b 


1 8551 


1 8092 


1 7764 


I 7506 


1 7293 1 71 Jl 


290 


1 9242 


1 8465 


1 8006 


I 7676 


1 7117 


1 7203 1 7021 


280 


I 0155 


1 837S 


1 7917 


1 7586 


1 7327 


1 7112 


1 6928 


270 


1 9067 


1 8288 


1 7826 


1 7494 


1 7234 


I 7018 


1 6S33 


260 


1 8977 


1 8197 


17734 


1 7401 


17139 


I ()92I 


1 6735 


250 


1 8885 


1 8104 


1 7639 


1 7305 


1 7U41 


1 0822 


1 (.635 


240 


1 8791 


1 8009 


1 7543 


1 7206 


1 694 1 


1 0721 


1 (,532 


230 


1 8()9G 


1 7911 


' 1 7413 


17105 


1 6839 I 1)1)17 


1 6*26 1 


220 


1 8598 


1 7812 


1 7342 


1 7002 


1 6733 I ()5U f ) 


1 6316 


210 


1 8498 


17709 


1 7238 


1 6896 


1 0025 1 ( J98 


I 020 1 


200 


1 8396 


1 7605 


1 7131 


1 678(5 


I 6513 11.281 


[ OOS7 


190 


1 8291 


1 7497 


1 702] 


1 0673 


I 6397 I 6161) 


1 5' 106 


180 


1 8184 


1 7387 


1 6907 


1 6557 


I 1)278 1 604 1 


15811 1 


170 


1 8074 


17274 


1 6791 


1 6 1,37 


1 6155 | J 5917 1 5711 


160 


1 7961 


1 7157 


1 6670 


1 6313 


~1 6027"' 


1 .->78 r ) 1 5575 


150 


1 7845 


1 7037 


1 6546 | 


1 6184 


15894 156 IS 151-33 


140 


J 7726 


16913 


___ 


1 6050 


J 5755 1 550 1 


1 5285 


130 


1 7604 


1 6785 


1 6283 


15911 


15610 15354 15129 


120 


1 7478 


1 6652 


1 6144 


1 5766 


15458 15196 11901 


110 


1 7347 


1 6315 


1 5999 


1 5613 


] 5299 1 5029 1 1700 


100 


1 7213 


16372 


1 5848 


15454 


151 31 14852 14605 



II 



TABLES OF THE PROPERTIES OF STEAM 377 



in any Dnj Stale. 



Temp 
Cent 


Ptessuio m pounds pei sc[ inch 


160 

] 770S 


180 


200 

1 7511 


250 300 350 


400 


400 


1 7033 


1 7250 1 7034 1 0849 


I 0087 


350 


J 7381 


J 724.) 


I 7118 


1 0852 1 0030 I 0439 


I 0271 


300 


0952 


I 0810 


0082 


1 0400 J 0175 1 5970 


1 579S 


290 


0801 


1 0718 


0589 


1 0311 I 0077 1 5873 


1 5090 


280 


0707 


1 0023 


0193 


1 0212 1 5970 J 5771 


1 5589 


270 


0070 


1 0525 


0)91 


I Oil! 1 5872 1 5004 


I 5179 


260 


0~>72 


1 0425 


029 J 


1 0000 


1 5704 1 5553 


1 5305 


250 


0170 


1 0)22 


0188 


1 5898 1 5052 1 5438 


I 52 10 


240 


0)05 


1 02 Id 


0081 


1 5780 1 5537 1 5310 


1 5123 


230 


0277 


1 0100 


5909 


5071 15117 1 5194 


1 4995 


220 


1 01 10 


1 599 ) 


1 5854 


5551 1 5292 || I 5005 


I 4800 


210 

200 
190 


1 0031 

1 5912 
1 5789 


1 5S70 

1 5755 
5029 


1 5735 

15011 
1 5182 " 


5420 


1 5102 

I 5020 
1 1881 


1 478S 


1 1719 

J 1571 
1 4110 


5290 
5101 


180^ 


1 5001 


5197 


I 5 548 


50 M) 


1 1735 


44S2 


1 4251 


170 


1 5527 


5500 


J 5208 


1870 


1 1577 


J )IO 


J 4077 


160 


I 5387 


52 1 7 


1 5000 


1713 1 Nil 


41 10 


1 3891 


150 


1 5211 


5007 


1 4000 


1 1518 I 12') 5 


'!95 3 


1 309 ! 


140 


1 5088 


4908 


1 4743 


1 1373 , I 1047 


3753 


1 3482 


130 


1 1920 


1741 


1 4570 


I 1180 


I 38 17 


353!) 


] 325 ) 


120 


1 4755 


I 451) 1 


1 4380 


1 3980 


1 3031 


I )307 


] 3000 


110 


1 1571. 


I 1375 


1 1190 


1 3772 


1 3')99 


1 3050 


] 2737 


100 


I 1381 


1 U74 


1 3980 


J 3511 


I 3147 


1 2783 


1 2413 



378 



THERMODYNAMICS 



LAPP TI [ 



CD <M 00 1C 1C 



IO 1C 1C 1O 1C 



<N C3 O O 00 
00 C5 <N -H l> 
1C 1C O O O 



O O >C -H r- l> 

ic CD co o on ic 
ic ic ic o o co 



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INDEX 



J)Holuto /oio, 12, 35 
Lbsoiption bands in spectmm, 3(51 




Ldiabatie elasticity, 2Sb 

expansion, 21 

elloci on di\ness, SI 

ol a fluid, 81 

ol a peifoctc;a3, 22 

of slofiin, 328 
idi.itlioimal piocesfl, 129 

i" ' 'I' 264 

k, of, 59 

ui, isotlicimals of, 302 

sopnialionot thoconshdit ntsof 180 

stdiiddid, 229 

111 UltnilP, HI^CIK ItlLlVC, 5J 

Jouli '4, 5 r > 
Vn m,i< Innos Em i cl 1 141 Ml ion, 1 "is 
ullvi n, SI, 'm 
im.iual, K If , 200, 302, 310 iI2 

HIlclL'cllN ISOtllC Illials, 2')') 

nuindmir to ( '.il- 
li mini s equation ot siati , 12 5 

iiimioma ilis<n plion in.ifliiiK, 11)1 

d.iln loi, 1 T) 

MSI ol 111 IcflL!{Uil.(l<H), l,}7 

LnllH^^^'^ ill) 

ilLinson 2 r >(i 

ilinnsplu 10 pirssuio of Rlandaid, S 

ilonni sd uc tmo, 3 r )S 

i\of,' idio's ld\\ , 235, 355 



, mean, 331 
C'aiboiuc acid, cidical teinpciatuie of,SO 

claLaloi, 145 

i&otheimal for, according 

to Glauauis, 317 

isothoiinnls of, 300 

inolccnlai vibi alien in, 

302 

specific heat of 245, 304 

use of, in lefngciation, 

138 

Garnot's cycle, conservation of entropy 
in, 46 

ot opeiaticma, 27 

icvoraod in lotugoiatmg 

inacliine, 138 

\sith a jtcitect gun foi 

\\oiUnifj; biiliHt.into, 30 

with slcani foi working 

siibslaiKo, S8 

('(HiiKlo inrlhod ot liqui f\ iiiL,' pases, IG'l 
Cli.iiiii Ii iistic i qiialion, 2 ( )0 

Callondai's, 31 

Clan <nis\ 315 

[)K tcuci's, 317 

of n )>(. i fee t ^nfl, 

200 

Van dei \Yauls 1 , 



lily's cm v( i, 185 
i( an do Jvochas, 22f> 
5oll ' i , 

5|i n ' ! ' 
Join, N , S5S 

?0lllllr, 310 

iniindai y cuivos 02 
!.i\lo's l.uv 13, 355 
SmiU'otUi, 320 



Clinks 1 la\\ 1 5 

(hail of utliopv and (i inpc lalim , 118 

- lolal heal, 121 

( hails ol pmpcidi's of Hinds, slopes of 

lini d in, 2SO 

( 'he inical i ontnu lion, 23(> 
machine, 15H Cla})eyion's equation, 115,283 

application of, to 

i linngcH ol fllalo, 1 1 (> 
Cldiido'n aj)|)aialiis loi complete jcctili 
( uf ion, 188 

liqiu ladion of an, 

178 

Sulish Association Committee on (,'laiisms, 50, 315 
gaseous ovploaions, 2 12, 2 I I, 241, 251 , ( 'hmsm.s c haiactcii d ie equal ion, .515 
257, 2(55 CloiiK-nt and Dcaoiinos, 204 

Jntish ThennalUnit, 9 r ' " r- '' '^ 229,244,251,259 

5nbblos, cquihbiiuin of, 350 i ' 

Juckingham, E , 277 ' .1,1 3, 319 

in alcani, 32!) 

'allendai, IT L , 12, 70, 72, 73, 70, 83, Coblcnt/, W W , %2 
102, 108, 127, 201, 257, 208, 315, 329, Ooofhcicnl of poifonnanLO, 2, 134, 13G 
335, 338, 350, 307 Coffoy still, 181 

'allondai and Nicholson, 207 Collected " ~ r ^ " r relations, 287 

'allondai's cliaiactonstic equation, 318 Combustic 

steam tables, 63, 340, 3(j7 Compound i ' I ! 220 



380 



INDEX 



Compressibility of a fluid, 2S6 
Compression, advantage of, in gas- 

eu "ines, 231 
Condensation in expanding steam, 207 

of water-vapour in an, 07 

Conservation of entropy in Cainol's 

cycle, 45 

m lefncei- 

atmg piocesa, 135 
Continuity of state, 304 
C'onveigent divergent nozzle, 194 
Cooling effect of tlnottlmg, 127, 270, 

296, 314, 318 

Con espon ding states, 311, 317 
Co- volume, 307, 319 

m steam, 328 
Cntical point 80, 304, 309 

in steam, 320 

on the I(j> chart, 123 

T</> chait, 120 

temperature of cat home acid, 80 

of hvdrogen, 81 

of steam SO 
Cuihs steam turbine, 220 
Curve of condition 218 

Cut oft, 97 

Cycle of operations, 4 

Dalby, W E , 125 2,57 

Dalton's principle, 05, 355 

David, W T 200 

Decrees of fieedom m gas molecules, 

201 358 
De Laval's nozzle, 193 

steam tin bine, 215 220 
Diatomic gases 261, 358 

specific heats of, 240 359 

molecules of h\diOLrcn, effect 

of extreme cold, 505 
Diesel engine, 234 

Dietoiici's characteiibtic equation, 317 
Discharge through nozzle, 194 

01 ifice, limitation of, 197 
Dissipation ot eneiLry, 50 
Disbocratron, 204 

Drops, foimation of, in supersaturated 

vapom, 85 344 

Diving air bv apphration of cold 07 
Drying pipes 100 

Drvness, change of, in achahatic ex- 
pansion, 93 

fraction, 70, 81 
Dust-free an, condensation in, 85 

Ebullition, 349 

Efficiency, conditions of maximum, 36 

of a heat-engine, 2 

of a perfect steam-engine, 90 

of a i e versible heat-engine, 29 

of Ranlane cycle, 99 

-intio, 110 

in a steam turbine, 210, 

219 



Elasticity of a fluid, 280 

Encigv clue to suiface tension, 341- 

internal, 5, Ifi 

molecular, 2(>U, 357 

of a gas 245, 357 

gas-engino mixture, 251 

of wbration in molecules, 301 
Engine with sepaiate oigans, 90 

workrng wi.thout expansion, 114 
Entiopy 44 

change of, in an in e\ersiblc opera- 

tion, 48 

of a fluid, 75 

ofwatei,334 

sum of, in a system of bodies, tO 

temperature chart fm steam 118 

temperatuie diagram for ic- 

fiigeiating cycle, 141 

temperatuie diagiam of perfect 

steam engine, 91 

-tempeiatiuc dmaiams, 40, 50 
Equilibrium of diops, 345 
Equipaitition of eneigj in tianslalion 

and rotation, 359 
Etlnl chloride, 138 
Eueken, A , 30(5 
Evaporation, 05 
Exhaust, 97 
Expansion cylinder, omission of, in rc- 



Explosiuii, ladialion in, 2bO 

External \\uik in foimatmn of st( am, (>0 

Extreme cold, pioduciion ot, 11)0 

Feed pump, \\oik ^pcnt in 97 
Fust la\\ ut thcimodv namics 5 
Flames experiments \\ilh, 2t)2 
Fluid cliaiiirtoiistiCKiiiatioiint 200,300 

functions of tlio state uf, 77, 20(> 
Fluids properties ot, 50 

Fon e, unit of 8 

Four stroke eye It 1 , 220 

Fiee eneig\ , 103 

Friction, effects ol, in jets and (mimics, 

209 

Function ft, 1 02 
Functions, Oilibs' theimodynamic, 103 

of the state of a llmd, 77, 200 

Gas, perfect, U, 290, 352 

theimometcr, 11 
Gas-constant, 238 
Gas-engine, 225 

mrxturo, energy of, 251 
Gases, combustion of, 235 

molecular energy of, 357 

theoiy of, 351 

piopertres of, ]3 

specific heats of, 239, 357 
Gauge-pressure, 05 

General theruiodynamic relations, 206 



INDEX 



381 



~!ibbs, Willaid, 72, 103 
ioodenough and Moshei, 1 J5 
Joiiclio, W J , 218 
Ji a mine-molecule, 237 
LJimdluy J H , 32!) 

Heat account in a mal piocess, 120 

dueel H -so of, to pioduce cold, 100, 

104, 1(>7 
~ l.ittnt, 00, G8 

muJnuuuil equivalent of, 10 

of loi mat ion oi steam, (57 

unit, of, ( ) 
Heal ih up, SI), 131 

application to the theory of 

juts, 192 

calculation of, 1UO 

tables, 110 
Heat engine, clcliiulion of, 1 

olluiency of, 2 
lfi.it ituiup, 1, 1,!3, 1.50 

I 1 ' ' i ot, 109 
I I 103 

Hi liiilioll/, It \on, 201 
HCMU'S iciiLticm tuibine, 221 
Hi use, W , 30I> 
ILinh uiuium, hcnclit <it, J 12 

iinpcnlaiu.p ot, in atoiiin 

tin buus, 223 

Ilollioin ind I kiimng, 2 r >3 
llnpkinvm, I! , 240 2 IS, 251, 2 r > r >, 250, 

2")7, 2 r iS, 2(>0 
Jl\dii>< a ciLtii.il luiipii ilme ol, hi 

isol In i tnal i of 301 

liquifm turn ill, 10 ( ) 

spriilit liuiL (if al \iiylott 

tompu.il nits 3l>0 

Iii, ( Hi < I dl pnssnii on tlie mulling 

I'Dinl ol, I 17 
ldi.il sli ,iiu Liiyino lollou mi; Cainots 

< \ i li , S,S 

liupuliil as, 2') r > 2 ( <) 
ImpiiUi I MI luiu , 220 
IIK omploli i \paiiHion, 112, 11! 
link pi inli ill \ iin.ililis, 20,S 
Indu <iloi diayiain, 7 
I ill i a i id spi 1 1 1 11 in ol awn, lilil 
liili'in.d i oinliiislion ongini', f>7 225 
lull inal cm i\, r >, 1C 

ill a, fluid, (){) 

of a gas, 215, 357 
Involution of c oolin^ elloc t, 277, Jld 
liiinusihlo ] n DCCHsos, 131 
Iriotliininal c iiivos, 298 

clawticity, 280 

ovpantuon, 22, 78 

of a poif ocL gas, 24 

lines on tlio piOHnui'o volume 

chauiam, 78 
Cm stoani 330 



Joans, J Jl , 306, 303 
Jonluu, G F , 145, 285 



Jet-pump, use of, in leliigciation, 156 
Jets, theoiy ot, 1U1 
Joly, J , 242 
Joule's au-ongine, 55 

icversed, 158 

equivalent, 10 

law, 15 

Joule Thomson cooling oflcct, 17, 127, 

270, 290, 314, 318 
Joule Thomson cooling cflect accoidmg 

to Callcndai's equation of state, 322 
Joule Thomson coolmi; oiled, use of, by 
Lindo, 171 

poiou9plugo\pciiment,74: 

Kelvin, Loul, 1(5, 39, 50, 117, 345 

Koycs and Biownloe, 145 

Kilo caloiy, 9 

Kinetic theoiy of gases, 300, 351 

Langon, 251 

Latent hodt, 00, (58 

Lmde, C, 171, 176, 181,27(5 

Liqnof.n. lion ol gaacs, Lindc'u method, 

170 

Liquid lilms, tension in, 313 
Ljungstiom tiiilnni 1 , 221 
Low pimsuii dkatn, IHC of, in attain 

tuiliinis, 223 
LuimiKi and Pim^sliiim, 215 

JMall.ud and Le ChaUlic-i, 251 

Mailin, II M , 2nS 

Matliot, 2M 

AIa\iinuin i UK H IK \ , i nndilions of ,!(> 

Ma\\\t H'sluiii llii unodj naniiL illations, 

271 
Mean tiio ]ialli, 351 

- tin imal unit, ( ), f U 
Mn InuiKul i(|in\ tilt nt, 10 
Milling poml, I'lU'i t of picssuie on, 

117 
Mi Uxlalili slali, 200 

and oquililiiiuin o\p.nnifin, 

205, 20S 

Mi\id nisis, picssiiu <if, 05 
Mi\luio ol a liquid \\ilh ils Riiliaalid 

viijinui, 73, 70, 2SJ 
Mol, 237 
IMoleuil.u eiiiigy ol a gas, 20 I , .557 

lhcoi\ of glides, 351 

\olocilus in LMHOS, 5 r >( 

woightH and \oliiinon, 235 
Alolounlos, encigy of, 357 

mimboi ol, in a cnliic cenli- 

mclio, 355 

in a gi<iuini(j- 

molLculc, 355 
Molhci, R, 121, 3K) 

cluut of total heat and ontiopy, 

124 

piG8surol25 

chaits for subafcaucoa used in 

lefugciation, 14.5 



382 



INDEX 



Monatomic gas, 261, 357 
Moss, H , 110 

Nernst, W , 205 

Newcomen, OS 

Newton's theory of transmission of 

sound, 29 i 
Nitiogen, isotheimals of, 301 

separation of, 181 
Nucleus, need of, in the formation of a 

diop, 344 

Oblique coordinates, use of in the /</> 

chart, 145 
Ols/sewski, 277 
Oigans ot a heat engine, 96 
Otto, 226 
Oxygen, separation of, 181 

Parsons' turbine, 216, 221 

Paitial pressures, G5 

Peake, A H , 320 

Perfect ind imperfect gases, 306 

ditteienlial, 270 

engine, enter loti ot, 29 

engine using regenerator, 50 

gas, 14, 23S, 325 

characteristic equation of, 

291) 

expansion of, 22, 24 

steam engine, 88 

efficiency of, 90 

eutiop} tempera- 

ture diagram of, 91 
Petrol engine, 225 
Phase, change of, 283 
Phases of a substance, 103 
Pier, 245 
Planck's constant, 3b i 

formula ior energj'- of vibration 

in molecules, 362 
Pol} atomic gases, specific heats, 364 

molecules, 359 
Portei, 3 IS 

Pound calory, 9 

Pressure, of a gas, explanation of, on 
the molecuJai theory, 352 

unit of, 8 

-volume diagram, G 
Preston, 295 

Radiation in explosions, 200 

from dames, 262 
Rankrue, 63, 70 

cycle, 98 

etticiency of, 102 

for steam in any state, 

104 

reveisrbilrty of, 109 
Rateau steam turbine, 220 
Rayleigh, Loul, 294 

Re action turbine, 221 
Rectification, 181 



Rectification, complete, 1SS 
^ ' '74, 120 

I. achme as a means oi 

warming, 1US 

ooLlIicicnlotnci- 

loimanco ol, 2 

drlimlion of, 1 
Re fiigtratron process, 133 
Regenerative air -engine, 53 

method ol producing ex- 

treme cold, 171 
Regenerator, Strrlrng's, 52 
Rcgnault, 14, 62, 241, 310 
Re-heat factor, 218 
Report ot Refrigeration Research Com- 

mrttee 145 
Reiersibrhty, 20 

conditions ol, 37 

the criterion ot per foe Iron, 

29 
Rc\eisiblc engine efficiency of, 34 

receiving licit at 

various temperatures, 42 

heat engine, 27 

reinitiating nu lime, lot 
Reynolds, Osbornc, 198 

Rontgcn, 295 
lUithertorcl, ,Srr E , 358 

Saturated steam, 61, 02 61, !(>S ,570 
relation ui pursuit' to 

tunpc laluu in, 33(> 
\apoui, 59 

Saturation due to cm \alutc, 3-)5 

ot an \\ Hli wati i-\ a |K nu, 66 
Scale ot temperature theimnd\ ii.unie 

12, 39 

he heel, K , 360 
Seay process, 103 

Second la\v ot tluimud\ nauncs, 26 
Mnt Ids, M C 1 , 361) 
Simple turbine, 215, 220 
bpccihcation of stale otaiij fluid, 77, 2d7 
Specific heat of \\alei, l>7 

\anation ol, with tcuip- 

ciatiiu, 213 

' ' 1!) 

in C'allcndai's equation of 

state, 32-1 

measurement oi value's ol, 

241 

of a gat), 17, 230, 357 

of a gas, influence ol 

molecular vibration, 363 

of gases on the molecular 

theorv 357 

of hydrogen at ver^ low 

tcuipeiaturcs, 365 

ratio of, 2 !, 275, 293, 3C7 
Stage efficiency in tuibuics, 218 
State, specification of, 77 

States of aggregation, 59 



INDEX 



383 



Steam, Callamlxi's lounulas foi, 327, Thomson, James, his ideal notheim.il, 



330, 338 

collcc Led fonnuliis 01, 338 

cntical pic'MHuio of, 320 

culical temper atuio of, SO 

cnliopy of, 37u 

Ruination o, imdoi constant 

pLessme, liO, 07 

jets, supcisaliualion m, 2(13 

piopeilics of, b2, 3o7 
saluialod, 01, t>2, M 

piopoitioa of, 30S 

apecilic ho it ot, 378 

Hiipeihoaled, 01 

tal.lea, 3 10, 307 

total heal of, 374 

tuibiuo, poiunmaneo of, 222 

Imhmes, compound, 216 
mmplo, 215 

typos ot, 220 

volume of, 372 

Mi un-i'iigiuo woikinLi, without coiu- 

piLisum, ') t 
Stilling, II , r >2 
Slodola, 203 
hut lion tt'inpoiadii o in a if. is engine, 

2/37 

Sndduv expansion, tltei I ul, S4 
SulpliimuiH at ul, L iS 
hup< Konlin^;, Sf), 201 1 
I^upi iluvitul \ iipoiu, f) ( ) 



tdl.il heat ol, 7 5 
\\-Ui i, U'l 

Sll[n is Li ill lllnll, S I 

ol Mliaiu disi li mu 

lioni a uu//li , 201 
Sin f.n <i It mion, 342 
N\\ uiti, 212 



Tables ot pi opi i tu s (it sti ,iiii, 3(>7 
TLIIIJK latino, HI iili s ul, 10 

of inviision ot iiHiliiiir 

t Hot t, 277 

tin i muil\ iiiimii HI ak ol, 

12, }<) 
I't in point in is in a u,an i n 

2 r )5 

Tension of liijind dim, 3 13 
Thtimal unil used in Calluidars 



units, ') 
r l i ni i i 
l.i 
Thciitnodyiuiiuu 
' 



T 

'I 



ou ec lion ot Iho 



eulialH, 101 
liom, 2d() 

(olIc('kul,2H7 
)l toiupoialuio, 
12, 3') 
suiliico, 2(58 

HC10I1CO Of, 1 

117, 304, 316 



305, 317 
Thomson, fcjn J J , 345 
Thom-,011, W, Loul Kelvin, 1C, 3'), CO, 

117,345 
Thiollling caloiinioU'i, 128 

cooling cAecfc of, 127, 27d, 

29t> 

pioccsa, 74 

Total heat, constancy of m a lluolllm 
piocoas, 74 

ol a iluid, 70, 72 

of watei, 334 
Tnalnnno gas, 2(>1 

molet'ulca, 350 
Tiiplu point, 1 IS 
Tm hint s, tvpis ol, 220 
Tuihn'mco, oilccti oi, 251) 
T\vo-sliol\.o cycln, 22(j 

Unit ol toicc, 8 

lital, ') 

piestuuo, 8 
woi k S 

Unit sisttil t-xpaiiHioii, 270 

Van dui Waals 1 chai.u tonstic equation, 

30(> 
Ihorii y tif toi itsponding 

4lale4 311 
V r .i[ioui < ompi tssiiin lolii^;* i at nig ninch- 

IIH, MS 
\.ipoui jiussuio ovoi a cin\L.d autfaco, 

3l r > 

\ i loi il\ r ut nuan s(|uiiio, i r ) i, 3f>l) 
\ ilu ilion ul itdins in mul(< uli M, i()0 
\ oliiiiK'ti IL spi i iii< h< ats, 23 f > 



Wal< i at sut m uf ion pussuir, piopulit'H 
ol, 3d 1 ) 

S|l( I ill! Ill III ol, ()7, I {"> 

supi i In al mi' ul, 3 IM 

int.ii iic'U mill i ni mp^ oi, ,n i 

\\ r uli i \ iipiiin n 1 1 ii>t i al inu mat hme, lf>f) 
Wall .lanus, ',S 
Wat I'M iiidii.itiu, 7 
\\ r (iglil, vaiulion ol, with liititudi, 8 
c ylnidi i , \Vi t nti urn, 7( 

Wilson, (' T R , .S, r >, 200, 207, 311, 3J' 
Wimpcin II U , 2f)0 
\Vilkownki, (03 

Woik dono by change of volume, (> 
done in adi.ihatio ovpaiiMinn, S(> 

uiul ot, H 
Working HnbHtance, 2 

eyelo ol opuiatiuiiH 

ol, I 



f, S3'dncy, 313 



CPHS, 137 



Xouuoi, 03, 82 

Zolly stoam turbmo, 220 



CAMBRIDGE PRINTPD B\ 

J B PEACE, M A , 
AT THF UNIVERSITY PRESS 






\JMBRIDGE PRINTED BY 

J B PEACE, M A , 
AT THE UNIVERSITY PRESS